http://cantorsattic.info/api.php?action=feedcontributions&user=Wabb2t&feedformat=atomCantor's Attic - User contributions [en]2022-08-19T19:30:43ZUser contributionsMediaWiki 1.24.4http://cantorsattic.info/index.php?title=Constructible_universe&diff=2174Constructible universe2017-11-18T08:31:31Z<p>Wabb2t: </p>
<hr />
<div>[[Category:Constructibility]]<br />
The Constructible universe (denoted $L$) was invented by Kurt Gödel as a transitive inner model of [[GCH]] + [[ZFC]] (assuming the consistency of ZFC) showing that ZFC cannot disprove GCH. It was then shown to be an important model of ZFC for its satisfying of other axioms, thus making them consistent with ZFC. The idea is that $L$ is built up by ranks like $V$. $L_0$ is the empty set, and $L_{\alpha+1}$ is the set of all easily definable subsets of $L_\alpha$. The assumption that $V=L$ (also known as the '''Axiom of constructibility''') is undecidable from ZFC, and implies many axioms which are consistent with ZFC to be true. A set $X$ is '''constructible''' iff $X\in L$. $V=L$ iff every set is constructible.<br />
<br />
== Definition ==<br />
<br />
$\mathrm{def}(X)$ is the set of all "easily definable" subsets of $X$ (specifically the $\Delta_0$ definable subsets). More specifically, a subset $x$ of $X$ is in $\mathrm{def}(X)$ iff there is a first-order formula $\varphi$ and $v_0,v_1...v_n\in X$ such that $x=\{y\in X:\varphi^X[y,v_0,v_1...v_n]\}$. Then, $L_\alpha$ and $L$ are defined as follows:<br />
<br />
*$L_0=\emptyset$<br />
*$L_{\alpha+1}=\mathrm{def}(L_\alpha)$<br />
*$L_\beta=\bigcup_{\alpha<\beta} L_\alpha$ if $\beta$ is a limit ordinal<br />
*$L=\bigcup_{\alpha\in\mathrm{Ord}} L_\alpha$<br />
<br />
=== The Relativized Constructible Universes $L_\alpha(W)$ and $L_\alpha[W]$ ===<br />
<br />
$L_\alpha(W)$ for a class $W$ is defined the same way except $L_0(W)=\text{TC}(\{W\})$ (the transitive closure of $\{W\}$). $L_\alpha[W]$ for a class $W$ is defined in the same way except using $\mathrm{def}_W(X)$, where $\mathrm{def}_W(X)$ is the set of all $x\subseteq X$ such that there is a first-order formula $\varphi$ and $v_0,v_1...v_n\in X$ such that $x=\{y\in X:\varphi^X[y,W,v_0,v_1...v_n]\}$ (because the relativization of $\varphi$ to $X$ is used and $\langle X,\in\rangle$ is not used, this definition makes sense even when $W$ is not in $X$).<br />
<br />
== The Difference Between $L_\alpha$ and $V_\alpha$ ==<br />
<br />
For $\alpha\leq\omega$, $L_\alpha=V_\alpha$. However, $|L_{\omega+\alpha}|=\aleph_0 + |\alpha|$ whilst $|V_{\omega+\alpha}|=\beth_\alpha$. Unless $\alpha$ is a [[Beth|$\beth$-fixed point]] or $\omega$, $|L_{\omega+\alpha}|<|V_{\omega+\alpha}|$. Although $L_\alpha$ is quite small compared to $V_\alpha$, $L$ is a tall model, meaning $L$ contains every ordinal. In fact, $V_\alpha\cap\mathrm{Ord}=L_\alpha\cap\mathrm{Ord}=\alpha$, so the ordinals in $V_\alpha$ are precisely those in $L_\alpha$. <br />
<br />
If [[Zero sharp|$0^{\#}$]] exists, then every uncountable cardinal $\kappa$ has $L\models\kappa\;\mathrm{is}\;\mathrm{totally}\;\mathrm{ineffible}$ (and therefore the smallest actually [[Ramsey|totally ineffible]] cardinal $\lambda$ has many more large cardinal properties in $L$). <br />
<br />
However, if $\kappa$ is [[inaccessible]] and $V=L$, then $V_\kappa=L_\kappa$. Furthermore, $V_\kappa\models (V=L)$. In the case where $V\neq L$, it is still true that $V_\kappa^L=L_\kappa$, although $V_\kappa^L$ will not be $V_\kappa$. In fact, $\mathcal{P}(\omega)\not\in V_\kappa^L$ if $0^{\#}$ exists.<br />
== Statements True in $L$ ==<br />
<br />
Here is a list of statements true in $L$:<br />
<br />
*[[ZFC]] (and therefore the Axiom of Choice)<br />
*[[GCH]]<br />
*$V=L$ (and therefore $V$ $=$ [[HOD Conjecture | $HOD$]])<br />
*The Diamond Principle <br />
*The Clubsuit Principle<br />
*The Falsity of Suslin's Hypothesis<br />
<br />
== Using Other Logic Systems than First-order Logic ==<br />
<br />
When using second order logic in the definition of $\mathrm{def}$, the new hierarchy is called $L_\alpha^{II}$. Interestingly, $L^{II}=HOD$. When using $\mathcal{L}_{\kappa,\kappa}$, the hierarchy is called $L_\alpha^{\mathcal{L}_{\kappa,\kappa}}$, and $L\subseteq L^{\mathcal{L}_{\kappa,\kappa}}\subseteq L(V_\kappa)$. Therefore, $V\neq L^{\mathcal{L}_{\kappa,\kappa}}$ iff $V\neq L$. Finally, when using $\mathcal{L}_{\infty,\infty}$, it turns out that the result is $V$.<br />
<br />
Chang's Model is $L^{\mathcal{L}_{\omega_1,\omega_1}}$. Chang proved that $L^{\mathcal{L}_{\kappa,\kappa}}$ is the smallest inner model of ZFC closed under sequences of length $<\kappa$. <br />
<br />
== References ==<br />
*Jech, ''Thomas J. Set Theory'' (The 3rd Millennium Ed.). Springer, 2003.<br />
*user46667, ''Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)'', URL (version: 2014-03-17): https://mathoverflow.net/q/156940<br />
*Chang, C. C. (1971), "Sets Constructible Using $\mathcal{L}_{\kappa,\kappa}$", ''Axiomatic Set Theory'', Proc. Sympos. Pure Math., XIII, Part I, Providence, R.I.: Amer. Math. Soc., pp. 1–8</div>Wabb2thttp://cantorsattic.info/index.php?title=Projective&diff=2098Projective2017-11-11T22:30:21Z<p>Wabb2t: /* Projective determinacy from large cardinals */</p>
<hr />
<div>{{DISPLAYTITLE: Projective sets and determinacy}}<br />
We say that $\Gamma$ is a ''pointclass'' if it is a collection of subsets of a [[:wikipedia:Polish space|Polish space]]. <br />
The '''lightface and bolface projective hierarchies''' are hierarchies of pointclasses of some Polish space $X$ defined by repeated applications of projections and complementations from either recursively enumerable or closed sets respectively.<br />
<br />
''Most results in this article can be found in <cite>Jech2003:SetTheory</cite> and <cite>Kanamori2009:HigherInfinite</cite> unless indicated otherwise.''<br />
<br />
== Definitions ==<br />
<br />
The following definitions are made by taking $X=\omega^\omega$, the ''Baire space'', i.e. the set of all functions $f:\mathbb{N}\to\mathbb{N}$. We will identify its members with the corresponding real numbers under some fixed bijection between $\mathbb{R}$ and $\omega^\omega$. The definitions presented here can be easily extended to other Polish spaces than the Baire space.<br />
<br />
Let $\mathbf{\Sigma}^0_1$ be the pointclass that contains all open subsets of the Polish space $\omega^\omega$. Let $\mathbf{\Pi}^0_1$ be the pointclass containing the complements of the $\mathbf{\Sigma}^0_1$ sets.<br />
<br />
We define the '''boldface projective pointclasses''' $\mathbf{\Sigma}^1_n$, $\mathbf{\Pi}^1_n$ and $\mathbf{\Delta}^1_n$ the following way:<br />
# $\mathbf{\Sigma}^1_1$ contains all the images of $\mathbf{\Pi}^0_1$ sets by continuous functions; its members are called the ''analytic sets''.<br />
# Now, for all $n$, define $\mathbf{\Pi}^1_n$ to be the set of the complements of the $\mathbf{\Sigma}^1_n$ sets; the members of $\mathbf{\Pi}^1_1$ are called the ''coanalytic sets''.<br />
# For all $n$, define $\mathbf{\Sigma}^1_{n+1}$ to be the set of the images of $\mathbf{\Pi}^1_n$ sets by continuous functions.<br />
# Finally, let $\mathbf{\Delta}^1_n=\mathbf{\Sigma}^1_n\cap\mathbf{\Pi}^1_n$. The members of $\mathbf{\Delta}^1_1$ are the ''Borel'' sets.<br />
<br />
The '''relativized lightface projective pointclasses''' $\Sigma^1_n(a)$, $\Pi^1_n(a)$ and $\Delta^1_n(a)$ (for $a\in\omega^\omega$) are defined similarly except that $\Sigma^1_1(a)$ is defined as the set of all $A\subseteq\omega^\omega$ such that $A=\{x\in\omega^\omega:\exists y\in\omega^\omega$ $\exists n\in\omega$ $R(x\restriction n,y\restriction n,a\restriction n)\}$, that is, $A$ is recursively definable by a formula with only existential quantifiers ranging on members of $\omega^\omega$ or on $\omega$ and whose only parameter is $a$.<br />
<br />
The (non-relativized) '''lightface projective classes''', also known as ''analytical pointclasses'', are the special cases $\Sigma^1_n$, $\Pi^1_n$ and $\Delta^1_n$ of relativized lightface projective pointclasses where $a=\empty$. Let $\Sigma^0_1$ be the pointclass of all ''recursively enumerable'' sets, i.e. the sets $A$ such there exists a recursive relation $R$ such that $A=\{x\in\omega^\omega:\exists n\in\omega$ $R(x\restriction n)\}$, and $\Pi^0_1$ contain the completements of $\Sigma^0_1$ sets. Then the $\Sigma^1_1$ sets are precisely the projections of $\Pi^0_1$ sets.<br />
<br />
Given an arbitrary pointclass $\Gamma$, define $\neg\Gamma$ as the set of the complements of $\Gamma$'s elements, for example $\Pi^1_n(a)=\neg\Sigma^1_n(a)$. Also let $\Delta_\Gamma=\Gamma\cap\neg\Gamma$, for example $\Delta^1_n(a)=\Delta_{\Pi^1_n(a)}=\Delta_{\Sigma^1_n(a)}$.<br />
<br />
== Properties ==<br />
<br />
Every $\mathbf{\Sigma}^1_n$ set is $\Sigma^1_n(a)$ for some $a\in\omega^\omega$, in fact $\mathbf{\Sigma}^1_n=\bigcup_{a\in\omega^\omega}\Sigma^1_n(a)$. A similar statement holds for $\mathbf{\Pi}^1_n$ sets and $\mathbf{\Delta}^1_n$ sets. This means the boldface projective sets are precisely the set definable using only real and arithmetical quantifiers and real parameters.<br />
<br />
The following statements also holds when replacing relativized lightface pointclasses by their boldface counterparts:<br />
* If $A$ and $B$ are $\Sigma^1_n(a)$ relations, then so are $\exists x\in\omega^\omega$ $A$, $A\land B$, $A\lor B$, $\exists n\in\omega$ $A$ and $\forall n\in\omega$ $A$.<br />
* If $A$ and $B$ are $\Pi^1_n(a)$ relations, then so are $\forall x\in\omega^\omega$ $A$, $A\land B$, $A\lor B$, $\exists n\in\omega$ $A$ and $\forall n\in\omega$ $A$.<br />
* If $A$ is a $\Sigma^1_n(a)$ relation then $\neg A$ is a $\Pi^1_n(a)$ relation. If $A$ is $\Pi^1_n(a)$ then $\neg A$ is $\Sigma^1_n(a)$.<br />
* If $A$ is a $\Sigma^1_n(a)$ relation and $B$ is a $\Pi^1_n(a)$ relation, then $A\Rightarrow B$ is a $\Pi^1_n(a)$ relation.<br />
* If $A$ is a $\Pi^1_n(a)$ relation and $B$ is a $\Sigma^1_n(a)$ relation, then $A\Rightarrow B$ is a $\Sigma^1_n(a)$ relation.<br />
* If $A$ and $B$ are $\Delta^1_n(a)$, then so are $\neg A$, $A\land B$, $A\lor B$, $A\Rightarrow B$, $A\Leftrightarrow B$, $\exists n\in\omega$ $A$, $\forall n\in\omega$ $A$.<br />
* $\Delta^1_n(a)\subsetneq\Sigma^1_n(a)\subsetneq\Delta^1_{n+1}(a)$<br />
* $\Delta^1_n(a)\subsetneq\Pi^1_n(a)\subsetneq\Delta^1_{n+1}(a)$<br />
<br />
Sierpinski showed that every $\mathbf{\Sigma}^1_2$ set of reals is the union of $\aleph_1$ Borel (=$\mathbf{\Delta}^1_1$) sets. It follows that every $\mathbf{\Sigma}^1_2$ set of reals is either (at most) countable or has the cardinality of the continuum.<br />
<br />
''Shoenfield’s absoluteness theorem'' is the statement that every $\Sigma^1_2(a)$ or $\Pi^1_2(a)$ relation is absolute for every inner model of $\text{ZF+DC}$ that contains $a$ (as an element). It follows that $\mathbf{\Sigma}^1_2$ and $\mathbf{\Pi}^1_2$ relations are absolute for $L$, and also that every $\Sigma^1_2(a)$ real (by taking $X=\omega$) is in $L[a]$, in particular every $\Sigma^1_2$ (or $\Pi^1_2$) real is constructible.<br />
The set of all constructible reals is $\Sigma^1_2$, and so is the canonical well-ordering $<_L$ of $L$. For $U$ a nonprincipal $\kappa$-complete [[ultrafilter]] on some [[measurable]] cardinal $\kappa$, then the collection of all sets of reals in $L[U]$ is $\Sigma^1_3$, and so is the canonical well-ordering $<_{L[U]}$ of $L[U]$. <br />
<br />
If [[zero sharp|$0^\#$]] exists then it is a $\Sigma^1_3$ real and the singleton $\{0^\#\}$ is a $\Pi^1_2$ set of reals. If for every real $a\in\omega^\omega$, the sharp $a^\#$ exists then every $\Sigma^1_3$ set of reals is the union of $\aleph_2$ Borel sets.<br />
<br />
== Regularity properties ==<br />
<br />
Let $A\subseteq(\omega^\omega)^k$ be a k-dimensional set of reals. We say that $A$ is ''null'' if it has [[:wikipedia:outer measure#Method I|outer measure]] 0. We say that $A$ is ''nowhere dense'' if its complement contains an open dense set, and that $A$ is ''meagre'' (or ''of first category'') if it is a countable union of nowhere dense set. Finally we say that $A$ is ''perfect'' if it has no isolated point.<br />
<br />
Then, we define the following ''regularity properties'':<br />
* $A$ is ''Lebesgue measurable'' if there exists a Borel set $B$ such that $A\Delta B$ is null.<br />
* $A$ has the ''Baire property'' if there exists an open set $B$ such that $A\Delta B$ is meagre.<br />
* $A$ has the ''perfect set property'' if it is either countable or has a perfect subset.<br />
Where $A\Delta B=(A\setminus B)\cup(B\setminus A)$ denotes symmetric difference. In $\text{ZFC}$ there exists Lebesgue non-measurable sets without the Baire property nor the perfect set property, but it is interesting to see which projective sets have those regularity properties.<br />
<br />
In $L$ there is a $\Delta^1_2$ set of reals that is not Lebesgue measurable and has no perfect subset. Also there is a $\Pi^1_1$ set of reals without the perfect set property.<br />
<br />
If every $\Sigma^1_3$ set of reals is Lebesgue measurable then $\aleph_1$ is [[inaccessible]] in $L$.<br />
<br />
If $A$ is a $\Sigma^1_2(a)$ set of reals and contains a real that is not in $L[a]$ then $A$ has the perfect set property. Note that every uncountable set with the perfect set property has the cardinality of the continuum.<br />
<br />
The following statements are equivalent:<br />
* For every real $a$, $\aleph_1^{L[a]}$ is countable.<br />
* Every $\mathbf{\Pi}^1_1$ set has the perfect set property.<br />
* Every $\mathbf{\Sigma}^1_2$ set has the perfect set property.<br />
<br />
If $E$ is a $\mathbf{\Pi}^1_1$ equivalence relation on $\omega^\omega$ then either $E$ has at most $\aleph_0$ equivalence classes or there exists a perfect set of mutually inequivalent reals. If $E$ is a $\mathbf{\Sigma}^1_1$ equivalence relation on $\omega^\omega$ then either $E$ has at most $\aleph_1$ equivalence classes or there exists a perfect set of mutually inequivalent reals.<br />
<br />
=== Prewellordering, scale and uniformization properties ===<br />
<br />
A ''norm'' on a set $A$ is a function $\varphi:A\to\text{Ord}$ from $A$ to the ordinals. A ''prewellordering'' is a relation $\preceq$ that is like a well-ordering except we do not require it to be reflexive or antisymmetric. If $\preceq$ is a prewellordering then the $a\equiv b\iff(a\preceq b\land b\preceq a)$ is an equivalence relation, and $\preceq$ is a well-ordering of the equivalence classes of $\equiv$. If $\varphi$ is a norm then $a\preceq_\varphi b\iff\varphi(a)\leq\varphi(b)$ is a prewellordering.<br />
<br />
A pointclass $\Gamma$ has the ''prewellordering property'' if every set $A$ in $\Gamma$ has a $\Gamma$-norm: a norm $\varphi:A\to\text{Ord}$, a $\Gamma$ relation $P(x,y)$ and a $\neg\Gamma$ relation $Q(x,y)$ such that for every $y\in A$ and all $x$: $x\in A\land\varphi(x)\leq\varphi(y)\iff P(x,y)\iff Q(x,y)$. The pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the prewellordering property.<br />
<br />
A ''scale'' on $A$ is a sequence of norms $\{\varphi_n:n\in\omega\}$ such that for every sequence of points $\{x_i:i\in\omega\}$, if for every $n$ the sequence $\{\varphi_n(x_i):n\in\omega\}$ is eventually constant with value $\alpha_n$, then $(lim_{i\to\omega}$ $x_i)\in A$ and for every $n$, $\varphi_n(lim_{i\to\omega}$ $x_i)\leq\alpha_n$. A pointclass $\Gamma$ has the ''scale property'' if for every set $A$ in $\Gamma$, there exists a $\Gamma$-scale for $A$: a scale $\{\varphi_n:n\in\omega\}$, a $\Gamma$ relation $P(n,x,y)$ and a $\neg\Gamma$ relation $Q(x,y)$ such that for every $n$, every $y\in A$ and all $x$: $x\in A\land\varphi_n(x)\leq\varphi_n(y)\iff P(n,x,y)\iff Q(n,x,y)$. Again, the pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the scale property.<br />
<br />
A set $A\subseteq\omega^\omega\times\omega^\omega$ is ''uniformized'' by a function $F$ if $dom(F)=\{x\in\omega^\omega:\exists y\in\omega^\omega$ $(x,y)\in A\}$ and $(x,F(x))\in A$ for all $x\in dom(F)$. A pointclass $\Gamma$ has the ''uniformization'' property if every set in $\Gamma$ can be uniformized by a function in $\Gamma$. The pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the uniformization property.<br />
<br />
=== Reduction and separation properties ===<br />
<br />
For any four sets $A$, $B$, $A'$ and $B'$, $(A',B')$ ''reduces'' $(A,B)$ if $A'\subseteq A$, $B'\subseteq B$, $A'\cup B'=A\cup B$, but $A'\cap B'=\empty$. Thus $A'$ and $B'$ partition $A\cup B$. A pointclass $\Gamma$ has the ''reduction property'' if for every $A$, $B$ in $\Gamma$ there exists $A'$, $B'$ in $\Gamma$ such that $(A',B')$ reduces $(A,B)$. $\Pi^1_1(a)$ has the reduction property for every $a\in\omega^\omega$.<br />
<br />
A pointclass $\Gamma$ has the ''separation property'' if for any disjoint subsets $A$, $B$ of $(\omega^\omega)^k$ for some $k$, if $A$ and $B$ are in $\Gamma$ then there is a $C$ in $\Delta_\Gamma$ such that $A\subseteq C$ and $B\cap C=\empty$. $\Sigma^1_1(a)$ has the separation property for every $a\in\omega^\omega$.<br />
<br />
If $\Gamma$ has the reduction property, then $\neg\Gamma$ has the separation property but not the reduction property. It is impossible for $\Gamma$ to have both the reduction and the separation properties. Every pointclass with the prewellordering property has the reduction property. Thus it is impossible for both a pointclass and its complement to have the prewellordering or scale properties.<br />
<br />
== Projective determinacy ==<br />
<br />
''See also: [[axiom of determinacy]]''<br />
<br />
''Determinacy'' is a kind of regularity property. For every set of reals $A$, the game $G_A$ is the infinite game of perfect information of length $\omega$ where both players constructs a sequence (i.e. a real) by playing elements of $\omega$, one after the other, such that the first player's goal is to have the constructed real be in $A$, and the second player's goal is to have the constructed real be in $A$'s complement. $A$ is ''determined'' if the game $G_A$ is determined, i.e. one of the two players have a winning strategy for $G_A$.<br />
<br />
Given a pointclass $\Gamma$, ''$\Gamma$-determinacy'' is the statement "every $A\in\Gamma$ is determined". $\Gamma$-determinacy and $\neg\Gamma$-determinacy are always equivalent. $\omega^\omega$-determinacy is the ''axiom of determinacy'' and is implied false by the [[axiom of choice]]. The '''axiom of projective determinacy''' ($\text{PD}$) is precisely $(\bigcup_{n\in\omega}\mathbf{\Sigma}^1_n)$-determinacy. Given some class $M$ (e.g. $\text{OD}$, $\text{L}(\mathbb{R})$, ...), ''$M$-determinacy'' is an abbreviation for $(M\cap\mathcal{P}(\omega^\omega))$-determinacy. $\text{L}(\mathbb{R})$-determinacy notably follows from large cardinal axioms, in particular the existence of infinitely many [[Woodin]] cardinals with a [[measurable]] above them all.<br />
<br />
Martin showed that $\text{ZFC}$ alone is sufficient to prove Borel determinacy (i.e. $\mathbf{\Delta}^1_1$-determinacy). However, for every $a\in\omega^\omega$, $\Sigma^1_1(a)$-determinacy is equivalent to "the sharp $a^\#$ exists", thus Borel determinacy is the best result possible in $\text{ZFC}$ alone. Analytic (i.e. $\mathbf{\Sigma^1_1}$) determinacy follows from the existence of a measurable cardinal, or even just of a [[Ramsey]] cardinal. Stronger forms of projective determinacies requires considerably stronger large cardinal axioms: for every $n$, $\mathbf{\Delta}^1_{n+1}$-determinacy implies the existence of an inner model with $n$ Woodin cardinals.<br />
<br />
Note that for every $n$, $\mathbf{\Sigma}^1_n$-determinacy is equivalent to $\mathbf{\Pi}^1_n$-determinacy. Furthertmore, under $\text{DC}$ (the ''[[:wikipedia:axiom of dependent choice|axiom of dependent choice]]'') for every $n\in\omega$, $\mathbf{\Delta}^1_{2n}$-determinacy is equivalent to $\mathbf{\Sigma}^1_{2n}$-determinacy ($\mathbf{\Pi}^1_{2n}$-determinacy)<br />
<br />
Assume $\mathbf{\Sigma}^1_n$ (or $\mathbf{\Pi}^1_n$) determinacy and that the axiom of choice holds for ''countable'' sets of reals (which follows from $\text{DC}$). Then every $\mathbf{\Sigma}^1_{n+1}$ set of reals is Lebesgue measurable, has the Baire property and has the perfect set property.<br />
<br />
Assume projective determinacy; then the following pointclasses have the reduction, prewellordering, scale and uniformization properties, for every $a\in\omega^\omega$: $\Pi^1_1(a), \Sigma^1_2(a), \Pi^1_3(a), ..., \Pi^1_{2n+1}(a), \Sigma^1_{2n+2}(a), ...$ This is known as the ''periodicity theorem''. On the other hand, if $L[U]$ contains every real for some nonprincipal $\kappa$-complete [[ultrafilter]] $U$ on a measurable cardinal $\kappa$, then every $\Sigma^1_n(a)$ has the reduction and prewellordering properties for $n\geq 2$ and every $a\in\omega^\omega$.<br />
<br />
=== Projective ordinals ===<br />
<br />
For every pointclass $\Gamma$, define $\delta_\Gamma$ as the supremum of the length of $\Gamma$ prewellorderings of $\omega^\omega$. We then define the ''projective ordinals'' to be $\delta^1_n=\delta_{\mathbf{\Sigma}^1_n}=\delta_{\mathbf{\Pi}^1_n}$. It can be shown without $\text{AD}$ that $\delta^1_1=\omega_1$ and that $\delta^1_2\leq\omega_2$. Under $\text{AD}$, each projective ordinal is a regular cardinal and the sequence $\{\delta^1_n:n\in\omega\}$ is a strictly increasing sequence of measurable cardinals, also $\delta^1_2=\omega_2$, $\delta^1_3=\omega_{\omega+1}$ and $\delta^1_4=\omega_{\omega+2}$. In general, $\delta^1_{2n+2}\leq(\delta^1_{2n+1})^{+}$. Under $\text{DC}$ this becomes an equality, also every $\delta^1_{2n+1}$ is the successor of a cardinal of cofinality $\omega$.<br />
<br />
Define $E:\omega\to\omega_1$ by recursion the following way: $E(0)=1$, $E(n+1)=\omega^{E(n)}$ (ordinal exponentiation). Then, under $\text{AD+DC}$, one have $\delta^1_{2n+3}=\omega_{E(2n+1)+1}$, also every $\delta^1_{2n+3}$ has the strong partition property $\delta^1_{2n+3}\to(\delta^1_{2n+3})^{\delta^1_{2n+3}}_\alpha$ for every $\alpha<\delta^1_{2n+3}$.<br />
<br />
Let's say a set of reals $A$ is $\gamma$-Borel (for a cardinal $\gamma$) if it is in the smallest collection of sets containing all closed sets of $(\omega^\omega)^k$ that is closed under complementations and unions of less than $\gamma$ sets. If $\gamma$ is not a cardinal then $A$ is $\gamma$-Borel if it is $\gamma^{+}$-Borel where $\gamma^{+}$ is the smallest cardinal >$\gamma$. Note that a set is Borel if and only if it is $\aleph_1$-Borel.<br />
<br />
Assume $\mathbf{\Delta}^1_{2n}$-determinacy; then a set of reals $A$ is $\mathbf{\Delta}^1_{2n+1}$ if and only if it is $\delta^1_{2n+1}$-Borel. Now, assume $\text{AD+DC}$; then a set $A$ is $\mathbf{\Sigma}^1_{2n+2}$ if and only if it is the union of $\delta^1_{2n+1}$-many sets that are $\mathbf{\Delta}^1_{2n+1}$.<br />
<br />
=== Projective determinacy from large cardinals ===<br />
<br />
Woodin showed that $\mathbf{\Pi}^1_{n+1}$-determinacy follows from the existence of $n$ [[Woodin]] cardinals with a measurable above them all, and projective determinacy thus follows from the existence of infinitely many Woodin cardinals. He also showed that $\mathbf{\Pi}^1_2$-determinacy is equivalent to "for all $x\in\mathbb{R}$, there is a countable ordinal $\delta$ such that $\delta$ is a Woodin cardinal in some inner model of $\text{ZFC}$ containing $x$, and that $\mathbf{\Delta}^1_2$-determinacy is equivalent to "for every $x\in\mathbb{R}$, there is an inner model M such that $x\in M$ and $M\models\text{ZFC}+$"there is a Woodin cardinal". <cite>KoellnerWoodin2010:LCFD</cite><br />
<br />
$\text{ZFC}$ + (lightface) $\Delta^1_2$-determinacy implies that there many $x$ such that $\text{HOD}^{\text{L}[x]}$ is a model of $\text{ZFC+}$"$\omega_2^{\text{L}[x]}$ is a Woodin cardinal".<br />
$\text{Z}_2$+$\Delta^1_2$-determinacy is conjectured to be equiconsistent with $\text{ZFC+}$"$\text{Ord}$ is Woodin", where "$\text{Ord}$ is Woodin" is expressed as an axiom scheme and $\text{Z}_2$ is [[:wikipedia:second-order arithmetic|second-order arithmetic]].<br />
$\text{Z}_3$+$\Delta^1_2$-determinacy is provably equiconsistent with $\text{NBG+}$"$\text{Ord}$ is Woodin" where $\text{NBG}$ is [[:wikipedia:Von Neumann–Bernays–Gödel set theory|Von Neumann–Bernays–Gödel set theory]] and $\text{Z}_3$ is third-order arithmetic.<br />
<br />
Gitik and Schindler showed that, in $\text{ZF}$, if $\aleph_\omega$ is a strong limit cardinal and $2^{\aleph_\omega}>\aleph_{\omega_1}$, then the axiom of projective determinacy holds. Also, if there is a singular cardinal of uncountable cofinality such that the sets of the cardinals below it such that the GCH holds is both [[stationary]] and costationary, then again the axiom of projective determinacy holds. It is not known whether these two results extends to $\text{L}(\mathbb{R})$-determinacy. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
Foreman, Magidor and Schindler showed that if there exists infinitely many cardinals $\delta$ above the continuum such that both $\delta$ and $\delta^{+}$ have the [[tree property]], then the axiom of projective determinacy holds. This hypothesis was shown to be consistent relative to the existence of infinitely many [[supercompact]] cardinals by James Cummings and Foreman. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Projective&diff=2097Projective2017-11-11T22:28:49Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: Projective sets and determinacy}}<br />
We say that $\Gamma$ is a ''pointclass'' if it is a collection of subsets of a [[:wikipedia:Polish space|Polish space]]. <br />
The '''lightface and bolface projective hierarchies''' are hierarchies of pointclasses of some Polish space $X$ defined by repeated applications of projections and complementations from either recursively enumerable or closed sets respectively.<br />
<br />
''Most results in this article can be found in <cite>Jech2003:SetTheory</cite> and <cite>Kanamori2009:HigherInfinite</cite> unless indicated otherwise.''<br />
<br />
== Definitions ==<br />
<br />
The following definitions are made by taking $X=\omega^\omega$, the ''Baire space'', i.e. the set of all functions $f:\mathbb{N}\to\mathbb{N}$. We will identify its members with the corresponding real numbers under some fixed bijection between $\mathbb{R}$ and $\omega^\omega$. The definitions presented here can be easily extended to other Polish spaces than the Baire space.<br />
<br />
Let $\mathbf{\Sigma}^0_1$ be the pointclass that contains all open subsets of the Polish space $\omega^\omega$. Let $\mathbf{\Pi}^0_1$ be the pointclass containing the complements of the $\mathbf{\Sigma}^0_1$ sets.<br />
<br />
We define the '''boldface projective pointclasses''' $\mathbf{\Sigma}^1_n$, $\mathbf{\Pi}^1_n$ and $\mathbf{\Delta}^1_n$ the following way:<br />
# $\mathbf{\Sigma}^1_1$ contains all the images of $\mathbf{\Pi}^0_1$ sets by continuous functions; its members are called the ''analytic sets''.<br />
# Now, for all $n$, define $\mathbf{\Pi}^1_n$ to be the set of the complements of the $\mathbf{\Sigma}^1_n$ sets; the members of $\mathbf{\Pi}^1_1$ are called the ''coanalytic sets''.<br />
# For all $n$, define $\mathbf{\Sigma}^1_{n+1}$ to be the set of the images of $\mathbf{\Pi}^1_n$ sets by continuous functions.<br />
# Finally, let $\mathbf{\Delta}^1_n=\mathbf{\Sigma}^1_n\cap\mathbf{\Pi}^1_n$. The members of $\mathbf{\Delta}^1_1$ are the ''Borel'' sets.<br />
<br />
The '''relativized lightface projective pointclasses''' $\Sigma^1_n(a)$, $\Pi^1_n(a)$ and $\Delta^1_n(a)$ (for $a\in\omega^\omega$) are defined similarly except that $\Sigma^1_1(a)$ is defined as the set of all $A\subseteq\omega^\omega$ such that $A=\{x\in\omega^\omega:\exists y\in\omega^\omega$ $\exists n\in\omega$ $R(x\restriction n,y\restriction n,a\restriction n)\}$, that is, $A$ is recursively definable by a formula with only existential quantifiers ranging on members of $\omega^\omega$ or on $\omega$ and whose only parameter is $a$.<br />
<br />
The (non-relativized) '''lightface projective classes''', also known as ''analytical pointclasses'', are the special cases $\Sigma^1_n$, $\Pi^1_n$ and $\Delta^1_n$ of relativized lightface projective pointclasses where $a=\empty$. Let $\Sigma^0_1$ be the pointclass of all ''recursively enumerable'' sets, i.e. the sets $A$ such there exists a recursive relation $R$ such that $A=\{x\in\omega^\omega:\exists n\in\omega$ $R(x\restriction n)\}$, and $\Pi^0_1$ contain the completements of $\Sigma^0_1$ sets. Then the $\Sigma^1_1$ sets are precisely the projections of $\Pi^0_1$ sets.<br />
<br />
Given an arbitrary pointclass $\Gamma$, define $\neg\Gamma$ as the set of the complements of $\Gamma$'s elements, for example $\Pi^1_n(a)=\neg\Sigma^1_n(a)$. Also let $\Delta_\Gamma=\Gamma\cap\neg\Gamma$, for example $\Delta^1_n(a)=\Delta_{\Pi^1_n(a)}=\Delta_{\Sigma^1_n(a)}$.<br />
<br />
== Properties ==<br />
<br />
Every $\mathbf{\Sigma}^1_n$ set is $\Sigma^1_n(a)$ for some $a\in\omega^\omega$, in fact $\mathbf{\Sigma}^1_n=\bigcup_{a\in\omega^\omega}\Sigma^1_n(a)$. A similar statement holds for $\mathbf{\Pi}^1_n$ sets and $\mathbf{\Delta}^1_n$ sets. This means the boldface projective sets are precisely the set definable using only real and arithmetical quantifiers and real parameters.<br />
<br />
The following statements also holds when replacing relativized lightface pointclasses by their boldface counterparts:<br />
* If $A$ and $B$ are $\Sigma^1_n(a)$ relations, then so are $\exists x\in\omega^\omega$ $A$, $A\land B$, $A\lor B$, $\exists n\in\omega$ $A$ and $\forall n\in\omega$ $A$.<br />
* If $A$ and $B$ are $\Pi^1_n(a)$ relations, then so are $\forall x\in\omega^\omega$ $A$, $A\land B$, $A\lor B$, $\exists n\in\omega$ $A$ and $\forall n\in\omega$ $A$.<br />
* If $A$ is a $\Sigma^1_n(a)$ relation then $\neg A$ is a $\Pi^1_n(a)$ relation. If $A$ is $\Pi^1_n(a)$ then $\neg A$ is $\Sigma^1_n(a)$.<br />
* If $A$ is a $\Sigma^1_n(a)$ relation and $B$ is a $\Pi^1_n(a)$ relation, then $A\Rightarrow B$ is a $\Pi^1_n(a)$ relation.<br />
* If $A$ is a $\Pi^1_n(a)$ relation and $B$ is a $\Sigma^1_n(a)$ relation, then $A\Rightarrow B$ is a $\Sigma^1_n(a)$ relation.<br />
* If $A$ and $B$ are $\Delta^1_n(a)$, then so are $\neg A$, $A\land B$, $A\lor B$, $A\Rightarrow B$, $A\Leftrightarrow B$, $\exists n\in\omega$ $A$, $\forall n\in\omega$ $A$.<br />
* $\Delta^1_n(a)\subsetneq\Sigma^1_n(a)\subsetneq\Delta^1_{n+1}(a)$<br />
* $\Delta^1_n(a)\subsetneq\Pi^1_n(a)\subsetneq\Delta^1_{n+1}(a)$<br />
<br />
Sierpinski showed that every $\mathbf{\Sigma}^1_2$ set of reals is the union of $\aleph_1$ Borel (=$\mathbf{\Delta}^1_1$) sets. It follows that every $\mathbf{\Sigma}^1_2$ set of reals is either (at most) countable or has the cardinality of the continuum.<br />
<br />
''Shoenfield’s absoluteness theorem'' is the statement that every $\Sigma^1_2(a)$ or $\Pi^1_2(a)$ relation is absolute for every inner model of $\text{ZF+DC}$ that contains $a$ (as an element). It follows that $\mathbf{\Sigma}^1_2$ and $\mathbf{\Pi}^1_2$ relations are absolute for $L$, and also that every $\Sigma^1_2(a)$ real (by taking $X=\omega$) is in $L[a]$, in particular every $\Sigma^1_2$ (or $\Pi^1_2$) real is constructible.<br />
The set of all constructible reals is $\Sigma^1_2$, and so is the canonical well-ordering $<_L$ of $L$. For $U$ a nonprincipal $\kappa$-complete [[ultrafilter]] on some [[measurable]] cardinal $\kappa$, then the collection of all sets of reals in $L[U]$ is $\Sigma^1_3$, and so is the canonical well-ordering $<_{L[U]}$ of $L[U]$. <br />
<br />
If [[zero sharp|$0^\#$]] exists then it is a $\Sigma^1_3$ real and the singleton $\{0^\#\}$ is a $\Pi^1_2$ set of reals. If for every real $a\in\omega^\omega$, the sharp $a^\#$ exists then every $\Sigma^1_3$ set of reals is the union of $\aleph_2$ Borel sets.<br />
<br />
== Regularity properties ==<br />
<br />
Let $A\subseteq(\omega^\omega)^k$ be a k-dimensional set of reals. We say that $A$ is ''null'' if it has [[:wikipedia:outer measure#Method I|outer measure]] 0. We say that $A$ is ''nowhere dense'' if its complement contains an open dense set, and that $A$ is ''meagre'' (or ''of first category'') if it is a countable union of nowhere dense set. Finally we say that $A$ is ''perfect'' if it has no isolated point.<br />
<br />
Then, we define the following ''regularity properties'':<br />
* $A$ is ''Lebesgue measurable'' if there exists a Borel set $B$ such that $A\Delta B$ is null.<br />
* $A$ has the ''Baire property'' if there exists an open set $B$ such that $A\Delta B$ is meagre.<br />
* $A$ has the ''perfect set property'' if it is either countable or has a perfect subset.<br />
Where $A\Delta B=(A\setminus B)\cup(B\setminus A)$ denotes symmetric difference. In $\text{ZFC}$ there exists Lebesgue non-measurable sets without the Baire property nor the perfect set property, but it is interesting to see which projective sets have those regularity properties.<br />
<br />
In $L$ there is a $\Delta^1_2$ set of reals that is not Lebesgue measurable and has no perfect subset. Also there is a $\Pi^1_1$ set of reals without the perfect set property.<br />
<br />
If every $\Sigma^1_3$ set of reals is Lebesgue measurable then $\aleph_1$ is [[inaccessible]] in $L$.<br />
<br />
If $A$ is a $\Sigma^1_2(a)$ set of reals and contains a real that is not in $L[a]$ then $A$ has the perfect set property. Note that every uncountable set with the perfect set property has the cardinality of the continuum.<br />
<br />
The following statements are equivalent:<br />
* For every real $a$, $\aleph_1^{L[a]}$ is countable.<br />
* Every $\mathbf{\Pi}^1_1$ set has the perfect set property.<br />
* Every $\mathbf{\Sigma}^1_2$ set has the perfect set property.<br />
<br />
If $E$ is a $\mathbf{\Pi}^1_1$ equivalence relation on $\omega^\omega$ then either $E$ has at most $\aleph_0$ equivalence classes or there exists a perfect set of mutually inequivalent reals. If $E$ is a $\mathbf{\Sigma}^1_1$ equivalence relation on $\omega^\omega$ then either $E$ has at most $\aleph_1$ equivalence classes or there exists a perfect set of mutually inequivalent reals.<br />
<br />
=== Prewellordering, scale and uniformization properties ===<br />
<br />
A ''norm'' on a set $A$ is a function $\varphi:A\to\text{Ord}$ from $A$ to the ordinals. A ''prewellordering'' is a relation $\preceq$ that is like a well-ordering except we do not require it to be reflexive or antisymmetric. If $\preceq$ is a prewellordering then the $a\equiv b\iff(a\preceq b\land b\preceq a)$ is an equivalence relation, and $\preceq$ is a well-ordering of the equivalence classes of $\equiv$. If $\varphi$ is a norm then $a\preceq_\varphi b\iff\varphi(a)\leq\varphi(b)$ is a prewellordering.<br />
<br />
A pointclass $\Gamma$ has the ''prewellordering property'' if every set $A$ in $\Gamma$ has a $\Gamma$-norm: a norm $\varphi:A\to\text{Ord}$, a $\Gamma$ relation $P(x,y)$ and a $\neg\Gamma$ relation $Q(x,y)$ such that for every $y\in A$ and all $x$: $x\in A\land\varphi(x)\leq\varphi(y)\iff P(x,y)\iff Q(x,y)$. The pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the prewellordering property.<br />
<br />
A ''scale'' on $A$ is a sequence of norms $\{\varphi_n:n\in\omega\}$ such that for every sequence of points $\{x_i:i\in\omega\}$, if for every $n$ the sequence $\{\varphi_n(x_i):n\in\omega\}$ is eventually constant with value $\alpha_n$, then $(lim_{i\to\omega}$ $x_i)\in A$ and for every $n$, $\varphi_n(lim_{i\to\omega}$ $x_i)\leq\alpha_n$. A pointclass $\Gamma$ has the ''scale property'' if for every set $A$ in $\Gamma$, there exists a $\Gamma$-scale for $A$: a scale $\{\varphi_n:n\in\omega\}$, a $\Gamma$ relation $P(n,x,y)$ and a $\neg\Gamma$ relation $Q(x,y)$ such that for every $n$, every $y\in A$ and all $x$: $x\in A\land\varphi_n(x)\leq\varphi_n(y)\iff P(n,x,y)\iff Q(n,x,y)$. Again, the pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the scale property.<br />
<br />
A set $A\subseteq\omega^\omega\times\omega^\omega$ is ''uniformized'' by a function $F$ if $dom(F)=\{x\in\omega^\omega:\exists y\in\omega^\omega$ $(x,y)\in A\}$ and $(x,F(x))\in A$ for all $x\in dom(F)$. A pointclass $\Gamma$ has the ''uniformization'' property if every set in $\Gamma$ can be uniformized by a function in $\Gamma$. The pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the uniformization property.<br />
<br />
=== Reduction and separation properties ===<br />
<br />
For any four sets $A$, $B$, $A'$ and $B'$, $(A',B')$ ''reduces'' $(A,B)$ if $A'\subseteq A$, $B'\subseteq B$, $A'\cup B'=A\cup B$, but $A'\cap B'=\empty$. Thus $A'$ and $B'$ partition $A\cup B$. A pointclass $\Gamma$ has the ''reduction property'' if for every $A$, $B$ in $\Gamma$ there exists $A'$, $B'$ in $\Gamma$ such that $(A',B')$ reduces $(A,B)$. $\Pi^1_1(a)$ has the reduction property for every $a\in\omega^\omega$.<br />
<br />
A pointclass $\Gamma$ has the ''separation property'' if for any disjoint subsets $A$, $B$ of $(\omega^\omega)^k$ for some $k$, if $A$ and $B$ are in $\Gamma$ then there is a $C$ in $\Delta_\Gamma$ such that $A\subseteq C$ and $B\cap C=\empty$. $\Sigma^1_1(a)$ has the separation property for every $a\in\omega^\omega$.<br />
<br />
If $\Gamma$ has the reduction property, then $\neg\Gamma$ has the separation property but not the reduction property. It is impossible for $\Gamma$ to have both the reduction and the separation properties. Every pointclass with the prewellordering property has the reduction property. Thus it is impossible for both a pointclass and its complement to have the prewellordering or scale properties.<br />
<br />
== Projective determinacy ==<br />
<br />
''See also: [[axiom of determinacy]]''<br />
<br />
''Determinacy'' is a kind of regularity property. For every set of reals $A$, the game $G_A$ is the infinite game of perfect information of length $\omega$ where both players constructs a sequence (i.e. a real) by playing elements of $\omega$, one after the other, such that the first player's goal is to have the constructed real be in $A$, and the second player's goal is to have the constructed real be in $A$'s complement. $A$ is ''determined'' if the game $G_A$ is determined, i.e. one of the two players have a winning strategy for $G_A$.<br />
<br />
Given a pointclass $\Gamma$, ''$\Gamma$-determinacy'' is the statement "every $A\in\Gamma$ is determined". $\Gamma$-determinacy and $\neg\Gamma$-determinacy are always equivalent. $\omega^\omega$-determinacy is the ''axiom of determinacy'' and is implied false by the [[axiom of choice]]. The '''axiom of projective determinacy''' ($\text{PD}$) is precisely $(\bigcup_{n\in\omega}\mathbf{\Sigma}^1_n)$-determinacy. Given some class $M$ (e.g. $\text{OD}$, $\text{L}(\mathbb{R})$, ...), ''$M$-determinacy'' is an abbreviation for $(M\cap\mathcal{P}(\omega^\omega))$-determinacy. $\text{L}(\mathbb{R})$-determinacy notably follows from large cardinal axioms, in particular the existence of infinitely many [[Woodin]] cardinals with a [[measurable]] above them all.<br />
<br />
Martin showed that $\text{ZFC}$ alone is sufficient to prove Borel determinacy (i.e. $\mathbf{\Delta}^1_1$-determinacy). However, for every $a\in\omega^\omega$, $\Sigma^1_1(a)$-determinacy is equivalent to "the sharp $a^\#$ exists", thus Borel determinacy is the best result possible in $\text{ZFC}$ alone. Analytic (i.e. $\mathbf{\Sigma^1_1}$) determinacy follows from the existence of a measurable cardinal, or even just of a [[Ramsey]] cardinal. Stronger forms of projective determinacies requires considerably stronger large cardinal axioms: for every $n$, $\mathbf{\Delta}^1_{n+1}$-determinacy implies the existence of an inner model with $n$ Woodin cardinals.<br />
<br />
Note that for every $n$, $\mathbf{\Sigma}^1_n$-determinacy is equivalent to $\mathbf{\Pi}^1_n$-determinacy. Furthertmore, under $\text{DC}$ (the ''[[:wikipedia:axiom of dependent choice|axiom of dependent choice]]'') for every $n\in\omega$, $\mathbf{\Delta}^1_{2n}$-determinacy is equivalent to $\mathbf{\Sigma}^1_{2n}$-determinacy ($\mathbf{\Pi}^1_{2n}$-determinacy)<br />
<br />
Assume $\mathbf{\Sigma}^1_n$ (or $\mathbf{\Pi}^1_n$) determinacy and that the axiom of choice holds for ''countable'' sets of reals (which follows from $\text{DC}$). Then every $\mathbf{\Sigma}^1_{n+1}$ set of reals is Lebesgue measurable, has the Baire property and has the perfect set property.<br />
<br />
Assume projective determinacy; then the following pointclasses have the reduction, prewellordering, scale and uniformization properties, for every $a\in\omega^\omega$: $\Pi^1_1(a), \Sigma^1_2(a), \Pi^1_3(a), ..., \Pi^1_{2n+1}(a), \Sigma^1_{2n+2}(a), ...$ This is known as the ''periodicity theorem''. On the other hand, if $L[U]$ contains every real for some nonprincipal $\kappa$-complete [[ultrafilter]] $U$ on a measurable cardinal $\kappa$, then every $\Sigma^1_n(a)$ has the reduction and prewellordering properties for $n\geq 2$ and every $a\in\omega^\omega$.<br />
<br />
=== Projective ordinals ===<br />
<br />
For every pointclass $\Gamma$, define $\delta_\Gamma$ as the supremum of the length of $\Gamma$ prewellorderings of $\omega^\omega$. We then define the ''projective ordinals'' to be $\delta^1_n=\delta_{\mathbf{\Sigma}^1_n}=\delta_{\mathbf{\Pi}^1_n}$. It can be shown without $\text{AD}$ that $\delta^1_1=\omega_1$ and that $\delta^1_2\leq\omega_2$. Under $\text{AD}$, each projective ordinal is a regular cardinal and the sequence $\{\delta^1_n:n\in\omega\}$ is a strictly increasing sequence of measurable cardinals, also $\delta^1_2=\omega_2$, $\delta^1_3=\omega_{\omega+1}$ and $\delta^1_4=\omega_{\omega+2}$. In general, $\delta^1_{2n+2}\leq(\delta^1_{2n+1})^{+}$. Under $\text{DC}$ this becomes an equality, also every $\delta^1_{2n+1}$ is the successor of a cardinal of cofinality $\omega$.<br />
<br />
Define $E:\omega\to\omega_1$ by recursion the following way: $E(0)=1$, $E(n+1)=\omega^{E(n)}$ (ordinal exponentiation). Then, under $\text{AD+DC}$, one have $\delta^1_{2n+3}=\omega_{E(2n+1)+1}$, also every $\delta^1_{2n+3}$ has the strong partition property $\delta^1_{2n+3}\to(\delta^1_{2n+3})^{\delta^1_{2n+3}}_\alpha$ for every $\alpha<\delta^1_{2n+3}$.<br />
<br />
Let's say a set of reals $A$ is $\gamma$-Borel (for a cardinal $\gamma$) if it is in the smallest collection of sets containing all closed sets of $(\omega^\omega)^k$ that is closed under complementations and unions of less than $\gamma$ sets. If $\gamma$ is not a cardinal then $A$ is $\gamma$-Borel if it is $\gamma^{+}$-Borel where $\gamma^{+}$ is the smallest cardinal >$\gamma$. Note that a set is Borel if and only if it is $\aleph_1$-Borel.<br />
<br />
Assume $\mathbf{\Delta}^1_{2n}$-determinacy; then a set of reals $A$ is $\mathbf{\Delta}^1_{2n+1}$ if and only if it is $\delta^1_{2n+1}$-Borel. Now, assume $\text{AD+DC}$; then a set $A$ is $\mathbf{\Sigma}^1_{2n+2}$ if and only if it is the union of $\delta^1_{2n+1}$-many sets that are $\mathbf{\Delta}^1_{2n+1}$.<br />
<br />
=== Projective determinacy from large cardinals ===<br />
<br />
Woodin showed that $\mathbf{\Pi}^1_{n+1}$-determinacy follows from the existence of $n$ [[Woodin]] cardinals with a measurable above them all, and projective determinacy thus follows from the existence of infinitely many Woodin cardinals. He also showed that $\mathbf{\Pi}^1_2$-determinacy is equivalent to "for all $x\in\mathbb{R}$, there is a countable ordinal $\delta$ such that $\delta$ is a Woodin cardinal in some inner model of $\text{ZFC}$ containing $x$, and that $\mathbf{\Delta}^1_2$-determinacy is equivalent to "for every $x\in\mathbb{R}$, there is an inner model M such that $x\in M$ and $M\models\text{ZFC}+$"there is a Woodin cardinal". <cite>KoellnerWoodin2010:LCFD</cite><br />
<br />
$\text{ZFC}$ + (lightface) $\Delta^1_2$-determinacy implies that there many $x$ such that $\text{HOD}^{\text{L}[x]}$ is a model of $\text{ZFC+}$"$\omega_2^{\text{L}[x]}$ is a Woodin cardinal".<br />
$\text{Z}_2$+$\Delta^1_2$-determinacy is conjectured to be equiconsistent with $\text{ZFC}$+"$\text{Ord}$ is Woodin", where "$\text{Ord}$ is Woodin" is expressed as an axiom scheme and $\text{Z}_2$ is [[:wikipedia:second-order arithmetic|second-order arithmetic]].<br />
$\text{Z}_3$+$\Delta^1_2$-determinacy is provably equiconsistent with $\text{NBG}+"$\text{Ord} is Woodin" where $\text{NBG}$ is [[:wikipedia:Von Neumann–Bernays–Gödel set theory|Von Neumann–Bernays–Gödel set theory]] and $\text{Z}_3$ is third-order arithmetic.<br />
<br />
Gitik and Schindler showed that, in $\text{ZF}$, if $\aleph_\omega$ is a strong limit cardinal and $2^{\aleph_\omega}>\aleph_{\omega_1}$, then the axiom of projective determinacy holds. Also, if there is a singular cardinal of uncountable cofinality such that the sets of the cardinals below it such that the GCH holds is both [[stationary]] and costationary, then again the axiom of projective determinacy holds. It is not known whether these two results extends to $\text{L}(\mathbb{R})$-determinacy. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
Foreman, Magidor and Schindler showed that if there exists infinitely many cardinals $\delta$ above the continuum such that both $\delta$ and $\delta^{+}$ have the [[tree property]], then the axiom of projective determinacy holds. This hypothesis was shown to be consistent relative to the existence of infinitely many [[supercompact]] cardinals by James Cummings and Foreman. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Huge&diff=2096Huge2017-11-11T22:22:04Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: Huge cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Critical points]]<br />
Huge cardinals (and their variants) were introduced by Kenneth Kunen in 1972 as a very large cardinal axiom. Kenneth Kunen first used them to prove that the consistency of the existence of a huge cardinal implies the consistency of $\text{ZFC}$+"there is a $\aleph_2$-[[filter|saturated ideal]] over $\omega_1$". <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
== Definitions ==<br />
<br />
Their formulation is similar to that of the formulation of [[superstrong]] cardinals. A huge cardinal is to a [[supercompact]] cardinal as a superstrong cardinal is to a [[strong]] cardinal, more precisely. The definition is part of a generalized phenomenon known as the "double helix", in which for some large cardinal properties $n$-$P_0$ and $n$-$P_1$, $n$-$P_0$ has less consistency strength than $n$-$P_1$, which has less consistency strength than $(n+1)$-$P_0$, and so on. This phenomenon is seen only around the [[n-fold variants|$n$-fold variants]] as of modern set theoretic concerns. <cite>Kentaro2007:DoubleHelix</cite><br />
<br />
Although they are very large, there is a first-order definition which is equivalent to $n$-hugeness, so the $\theta$-th $n$-huge cardinal is first-order definable whenever $\theta$ is first-order definable. This definition can be seen as a (very strong) strengthening of the first-order definition of [[measurable|measurability]].<br />
<br />
=== Elementary embedding definitions ===<br />
<br />
The elementary embedding definitions are somewhat standard. Let $j:V\rightarrow M$ be an [[elementary embedding]] with critical point $\kappa$ such that $M$ is a standard inner model of [[ZFC|$\text{ZFC}$]]. Then:<br />
<br />
*$\kappa$ is '''almost $n$-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length less than $\lambda$ (that is, $M^{<\lambda}\subseteq M$).<br />
*$\kappa$ is '''$n$-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length $\lambda$ ($M^\lambda\subseteq M$).<br />
*$\kappa$ is '''almost $n$-huge''' iff it is almost $n$-huge with target $\lambda$ for some $\lambda$.<br />
*$\kappa$ is '''$n$-huge''' iff it is $n$-huge with target $\lambda$ for some $\lambda$.<br />
*$\kappa$ is '''super almost $n$-huge''' iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is almost $n$-huge with target $\lambda$ (that is, the target can be made arbitrarily large).<br />
*$\kappa$ is '''super $n$-huge''' iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is $n$-huge with target $\lambda$.<br />
*$\kappa$ is '''almost huge''', '''huge''', '''super almost huge''', and '''superhuge''' iff it is '''almost $1$-huge''', '''$1$-huge''', etc. respectively.<br />
<br />
=== Ultrafilter definition ===<br />
<br />
The first-order definition of $n$-huge is somewhat similar to [[measurable|measurability]]. Specifically, $\kappa$ is measurable iff there is a nonprincipal $\kappa$-complete [[filter|ultrafilter]], $U$, over $\kappa$. A cardinal $\kappa$ is $n$-huge with target $\lambda$ iff there is a normal $\kappa$-complete ultrafilter, $U$, over $\mathcal{P}(\lambda)$, and cardinals $\kappa=\lambda_0<\lambda_1<\lambda_2...<\lambda_{n-1}<\lambda_n=\lambda$ such that:<br />
<br />
$$\forall i<n\forall x\subseteq\lambda(ot(x\cap\lambda_{i+1})=\lambda_i\rightarrow x\in U)$$<br />
<br />
Where $ot(X)$ is the [[Order-isomorphism|order-type]] of the poset $(X,\in)$. <cite>Kanamori2009:HigherInfinite</cite> This definition is, more intuitively, making $U$ very large, like most ultrafilter characterizations of large cardinals ([[supercompact]], [[strongly compact]], etc.).<br />
$\kappa$ is then super $n$-huge if for all ordinals $\theta$ there is a $\lambda>\theta$ such that $\kappa$ is $n$-huge with target $\lambda$, i.e. $\lambda_n$ can be made arbitrarily large. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
If $j:V\to M$ is such that $M^{j^n(\kappa)}\subseteq M$ (i.e. $j$ witnesses $n$-hugeness) then there is a ultrafilter $U$ as above such that, for all $k\leq n$, $\lambda_k = j^k(\kappa)$, i.e. it is not only $\lambda=\lambda_n$ that is an iterate of $\kappa$ by $j$; all members of the $\lambda_k$ sequence are.<br />
<br />
== Consistency strength and size ==<br />
<br />
Hugeness exhibits a phenomenon associated with similarly defined large cardinals (the [[n-fold variants|$n$-fold variants]]) known as the ''double helix''. This phenomenon is when for one $n$-fold variant, letting a cardinal be called $n$-$P_0$ iff it has the property, and another variant, $n$-$P_1$, $n$-$P_0$ is weaker than $n$-$P_1$, which is weaker than $(n+1)$-$P_0$. <cite>Kentaro2007:DoubleHelix</cite> In the consistency strength hierarchy, here is where these lay (top being weakest):<br />
<br />
* [[measurable]] = $0$-[[superstrong]] = almost $0$-huge = super almost $0$-huge = $0$-huge = super $0$-huge <br />
* $n$-superstrong<br />
* $n$-fold supercompact<br />
* $(n+1)$-fold strong, $n$-fold extendible<br />
* $(n+1)$-fold Woodin, $n$-fold Vopěnka<br />
* $(n+1)$-fold Shelah<br />
* almost $n$-huge<br />
* super almost $n$-huge<br />
* $n$-huge<br />
* super $n$-huge<br />
* $(n+1)$-superstrong<br />
<br />
All huge variants lay at the top of the double helix restricted to some [[Omega|natural number]] $n$, although each are bested by [[rank-into-rank|I3]] cardinals (the [[elementary embedding|critical points]] of the I3 elementary embeddings). In fact, every I3 is preceeded by a stationary set of $n$-huge cardinals, for all $n$. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
Similarly, every huge cardinal $\kappa$ is almost huge, and there is a normal measure over $\kappa$ which contains every almost huge cardinal $\lambda<\kappa$. Every superhuge cardinal $\kappa$ is [[extendible]] and there is a normal measure over $\kappa$ which contains every extendible cardinal $\lambda<\kappa$. Every $(n+1)$-huge cardinal $\kappa$ has a normal measure which contains every cardinal $\lambda$ such that $V_\kappa\models$"$\lambda$ is super $n$-huge". <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
In terms of size, however, the least $n$-huge cardinal is smaller than the least [[supercompact]] cardinal. Assuming both exist, for any $\kappa$ which is supercompact and has an $n$-huge cardinal above it, there are $\kappa$ many $n$-huge cardinals less than $\kappa$. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
Every $n$-huge cardinal is $m$-huge for every $m\leq n$. Similarly with almost $n$-hugeness, super $n$-hugeness, and super almost $n$-hugeness. Every almost huge cardinal is [[Vopenka|Vopěnka]] (therefore the consistency of the existence of an almost-huge cardinal implies the consistency of Vopěnka's principle). <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Huge&diff=2095Huge2017-11-11T22:21:31Z<p>Wabb2t: /* Ultrafilter definition */</p>
<hr />
<div>{{DISPLAYTITLE: Huge cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Critical points]]<br />
Huge cardinals (and their variants) were introduced by Kenneth Kunen in 1972 as a very large cardinal axiom. Kenneth Kunen first used them to prove that the consistency of the existence of a huge cardinal implies the consistency of $\text{ZFC}$+"there is a $\aleph_2$-[[filter|saturated ideal]] over $\omega_1$". <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
== Definitions ==<br />
<br />
Their formulation is similar to that of the formulation of [[superstrong]] cardinals. A huge cardinal is to a [[supercompact]] cardinal as a superstrong cardinal is to a [[strong]] cardinal, more precisely. The definition is part of a generalized phenomenon known as the "double helix", in which for some large cardinal properties $n$-$P_0$ and $n$-$P_1$, $n$-$P_0$ has less consistency strength than $n$-$P_1$, which has less consistency strength than $(n+1)$-$P_0$, and so on. This phenomenon is seen only around the [[n-fold variants|$n$-fold variants]] as of modern set theoretic concerns. <cite>Kentaro2007:DoubleHelix</cite><br />
<br />
Although they are very large, there is a first-order definition which is equivalent to $n$-hugeness, so the $\theta$-th $n$-huge cardinal is first-order definable whenever $\theta$ is first-order definable. This definition can be seen as a (very strong) strengthening of the first-order definition of [[measurable|measurability]].<br />
<br />
=== Elementary embedding definitions ===<br />
<br />
The elementary embedding definitions are somewhat standard. Let $j:V\rightarrow M$ be an [[elementary embedding]] with critical point $\kappa$ such that $M$ is a standard inner model of [[ZFC|$\text{ZFC}$]]. Then:<br />
<br />
*$\kappa$ is '''almost $n$-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length less than $\lambda$ (that is, $M^{<\lambda}\subset M$).<br />
*$\kappa$ is '''$n$-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length $\lambda$ ($M^\lambda\subset M$).<br />
*$\kappa$ is '''almost $n$-huge''' iff it is almost $n$-huge with target $\lambda$ for some $\lambda$.<br />
*$\kappa$ is '''$n$-huge''' iff it is $n$-huge with target $\lambda$ for some $\lambda$.<br />
*$\kappa$ is '''super almost $n$-huge''' iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is almost $n$-huge with target $\lambda$ (that is, the target can be made arbitrarily large).<br />
*$\kappa$ is '''super $n$-huge''' iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is $n$-huge with target $\lambda$.<br />
*$\kappa$ is '''almost huge''', '''huge''', '''super almost huge''', and '''superhuge''' iff it is '''almost $1$-huge''', '''$1$-huge''', etc. respectively.<br />
<br />
=== Ultrafilter definition ===<br />
<br />
The first-order definition of $n$-huge is somewhat similar to [[measurable|measurability]]. Specifically, $\kappa$ is measurable iff there is a nonprincipal $\kappa$-complete [[filter|ultrafilter]], $U$, over $\kappa$. A cardinal $\kappa$ is $n$-huge with target $\lambda$ iff there is a normal $\kappa$-complete ultrafilter, $U$, over $\mathcal{P}(\lambda)$, and cardinals $\kappa=\lambda_0<\lambda_1<\lambda_2...<\lambda_{n-1}<\lambda_n=\lambda$ such that:<br />
<br />
$$\forall i<n\forall x\subseteq\lambda(ot(x\cap\lambda_{i+1})=\lambda_i\rightarrow x\in U)$$<br />
<br />
Where $ot(X)$ is the [[Order-isomorphism|order-type]] of the poset $(X,\in)$. <cite>Kanamori2009:HigherInfinite</cite> This definition is, more intuitively, making $U$ very large, like most ultrafilter characterizations of large cardinals ([[supercompact]], [[strongly compact]], etc.).<br />
$\kappa$ is then super $n$-huge if for all ordinals $\theta$ there is a $\lambda>\theta$ such that $\kappa$ is $n$-huge with target $\lambda$, i.e. $\lambda_n$ can be made arbitrarily large. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
If $j:V\to M$ is such that $M^{j^n(\kappa)}\subseteq M$ (i.e. $j$ witnesses $n$-hugeness) then there is a ultrafilter $U$ as above such that, for all $k\leq n$, $\lambda_k = j^k(\kappa)$, i.e. it is not only $\lambda=\lambda_n$ that is an iterate of $\kappa$ by $j$; all members of the $\lambda_k$ sequence are.<br />
<br />
== Consistency strength and size ==<br />
<br />
Hugeness exhibits a phenomenon associated with similarly defined large cardinals (the [[n-fold variants|$n$-fold variants]]) known as the ''double helix''. This phenomenon is when for one $n$-fold variant, letting a cardinal be called $n$-$P_0$ iff it has the property, and another variant, $n$-$P_1$, $n$-$P_0$ is weaker than $n$-$P_1$, which is weaker than $(n+1)$-$P_0$. <cite>Kentaro2007:DoubleHelix</cite> In the consistency strength hierarchy, here is where these lay (top being weakest):<br />
<br />
* [[measurable]] = $0$-[[superstrong]] = almost $0$-huge = super almost $0$-huge = $0$-huge = super $0$-huge <br />
* $n$-superstrong<br />
* $n$-fold supercompact<br />
* $(n+1)$-fold strong, $n$-fold extendible<br />
* $(n+1)$-fold Woodin, $n$-fold Vopěnka<br />
* $(n+1)$-fold Shelah<br />
* almost $n$-huge<br />
* super almost $n$-huge<br />
* $n$-huge<br />
* super $n$-huge<br />
* $(n+1)$-superstrong<br />
<br />
All huge variants lay at the top of the double helix restricted to some [[Omega|natural number]] $n$, although each are bested by [[rank-into-rank|I3]] cardinals (the [[elementary embedding|critical points]] of the I3 elementary embeddings). In fact, every I3 is preceeded by a stationary set of $n$-huge cardinals, for all $n$. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
Similarly, every huge cardinal $\kappa$ is almost huge, and there is a normal measure over $\kappa$ which contains every almost huge cardinal $\lambda<\kappa$. Every superhuge cardinal $\kappa$ is [[extendible]] and there is a normal measure over $\kappa$ which contains every extendible cardinal $\lambda<\kappa$. Every $(n+1)$-huge cardinal $\kappa$ has a normal measure which contains every cardinal $\lambda$ such that $V_\kappa\models$"$\lambda$ is super $n$-huge". <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
In terms of size, however, the least $n$-huge cardinal is smaller than the least [[supercompact]] cardinal. Assuming both exist, for any $\kappa$ which is supercompact and has an $n$-huge cardinal above it, there are $\kappa$ many $n$-huge cardinals less than $\kappa$. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
Every $n$-huge cardinal is $m$-huge for every $m\leq n$. Similarly with almost $n$-hugeness, super $n$-hugeness, and super almost $n$-hugeness. Every almost huge cardinal is [[Vopenka|Vopěnka]] (therefore the consistency of the existence of an almost-huge cardinal implies the consistency of Vopěnka's principle). <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Huge&diff=2093Huge2017-11-11T22:13:55Z<p>Wabb2t: /* Ultrafilter definition */</p>
<hr />
<div>{{DISPLAYTITLE: Huge cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Critical points]]<br />
Huge cardinals (and their variants) were introduced by Kenneth Kunen in 1972 as a very large cardinal axiom. Kenneth Kunen first used them to prove that the consistency of the existence of a huge cardinal implies the consistency of $\text{ZFC}$+"there is a $\aleph_2$-[[filter|saturated ideal]] over $\omega_1$". <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
== Definitions ==<br />
<br />
Their formulation is similar to that of the formulation of [[superstrong]] cardinals. A huge cardinal is to a [[supercompact]] cardinal as a superstrong cardinal is to a [[strong]] cardinal, more precisely. The definition is part of a generalized phenomenon known as the "double helix", in which for some large cardinal properties $n$-$P_0$ and $n$-$P_1$, $n$-$P_0$ has less consistency strength than $n$-$P_1$, which has less consistency strength than $(n+1)$-$P_0$, and so on. This phenomenon is seen only around the [[n-fold variants|$n$-fold variants]] as of modern set theoretic concerns. <cite>Kentaro2007:DoubleHelix</cite><br />
<br />
Although they are very large, there is a first-order definition which is equivalent to $n$-hugeness, so the $\theta$-th $n$-huge cardinal is first-order definable whenever $\theta$ is first-order definable. This definition can be seen as a (very strong) strengthening of the first-order definition of [[measurable|measurability]].<br />
<br />
=== Elementary embedding definitions ===<br />
<br />
The elementary embedding definitions are somewhat standard. Let $j:V\rightarrow M$ be an [[elementary embedding]] with critical point $\kappa$ such that $M$ is a standard inner model of [[ZFC|$\text{ZFC}$]]. Then:<br />
<br />
*$\kappa$ is '''almost $n$-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length less than $\lambda$ (that is, $M^{<\lambda}\subset M$).<br />
*$\kappa$ is '''$n$-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length $\lambda$ ($M^\lambda\subset M$).<br />
*$\kappa$ is '''almost $n$-huge''' iff it is almost $n$-huge with target $\lambda$ for some $\lambda$.<br />
*$\kappa$ is '''$n$-huge''' iff it is $n$-huge with target $\lambda$ for some $\lambda$.<br />
*$\kappa$ is '''super almost $n$-huge''' iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is almost $n$-huge with target $\lambda$ (that is, the target can be made arbitrarily large).<br />
*$\kappa$ is '''super $n$-huge''' iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is $n$-huge with target $\lambda$.<br />
*$\kappa$ is '''almost huge''', '''huge''', '''super almost huge''', and '''superhuge''' iff it is '''almost $1$-huge''', '''$1$-huge''', etc. respectively.<br />
<br />
=== Ultrafilter definition ===<br />
<br />
The first-order definition of $n$-huge is somewhat similar to [[measurable|measurability]]. Specifically, $\kappa$ is measurable iff there is a nonprincipal $\kappa$-complete [[filter|ultrafilter]], $U$, over $\kappa$. A cardinal $\kappa$ is $n$-huge with target $\lambda$ iff there is a nonprincipal $\kappa$-complete ultrafilter, $U$, over $\mathcal{P}(\lambda)$, and cardinals $\kappa=\lambda_0<\lambda_1<\lambda_2...<\lambda_{n-1}<\lambda_n=\lambda$ such that:<br />
<br />
$$\forall i<n\forall x\subseteq\lambda(ot(x\cap\lambda_{i+1})=\lambda_i\rightarrow x\in U)$$<br />
<br />
Where $ot(X)$ is the [[Order-isomorphism|order-type]] of the poset $(X,\in)$. <cite>Kanamori2009:HigherInfinite</cite> This definition is, more intuitively, making $U$ very large, like most ultrafilter characterizations of large cardinals ([[supercompact]], [[strongly compact]], etc.).<br />
$\kappa$ is then super $n$-huge if for all ordinals $\theta$ there is a $\lambda>\theta$ such that $\kappa$ is $n$-huge with target $\lambda$, i.e. $\lambda_n$ can be made arbitrarily large. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
== Consistency strength and size ==<br />
<br />
Hugeness exhibits a phenomenon associated with similarly defined large cardinals (the [[n-fold variants|$n$-fold variants]]) known as the ''double helix''. This phenomenon is when for one $n$-fold variant, letting a cardinal be called $n$-$P_0$ iff it has the property, and another variant, $n$-$P_1$, $n$-$P_0$ is weaker than $n$-$P_1$, which is weaker than $(n+1)$-$P_0$. <cite>Kentaro2007:DoubleHelix</cite> In the consistency strength hierarchy, here is where these lay (top being weakest):<br />
<br />
* [[measurable]] = $0$-[[superstrong]] = almost $0$-huge = super almost $0$-huge = $0$-huge = super $0$-huge <br />
* $n$-superstrong<br />
* $n$-fold supercompact<br />
* $(n+1)$-fold strong, $n$-fold extendible<br />
* $(n+1)$-fold Woodin, $n$-fold Vopěnka<br />
* $(n+1)$-fold Shelah<br />
* almost $n$-huge<br />
* super almost $n$-huge<br />
* $n$-huge<br />
* super $n$-huge<br />
* $(n+1)$-superstrong<br />
<br />
All huge variants lay at the top of the double helix restricted to some [[Omega|natural number]] $n$, although each are bested by [[rank-into-rank|I3]] cardinals (the [[elementary embedding|critical points]] of the I3 elementary embeddings). In fact, every I3 is preceeded by a stationary set of $n$-huge cardinals, for all $n$. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
Similarly, every huge cardinal $\kappa$ is almost huge, and there is a normal measure over $\kappa$ which contains every almost huge cardinal $\lambda<\kappa$. Every superhuge cardinal $\kappa$ is [[extendible]] and there is a normal measure over $\kappa$ which contains every extendible cardinal $\lambda<\kappa$. Every $(n+1)$-huge cardinal $\kappa$ has a normal measure which contains every cardinal $\lambda$ such that $V_\kappa\models$"$\lambda$ is super $n$-huge". <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
In terms of size, however, the least $n$-huge cardinal is smaller than the least [[supercompact]] cardinal. Assuming both exist, for any $\kappa$ which is supercompact and has an $n$-huge cardinal above it, there are $\kappa$ many $n$-huge cardinals less than $\kappa$. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
Every $n$-huge cardinal is $m$-huge for every $m\leq n$. Similarly with almost $n$-hugeness, super $n$-hugeness, and super almost $n$-hugeness. Every almost huge cardinal is [[Vopenka|Vopěnka]] (therefore the consistency of the existence of an almost-huge cardinal implies the consistency of Vopěnka's principle). <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Huge&diff=2092Huge2017-11-11T22:12:11Z<p>Wabb2t: /* Ultrafilter definition */</p>
<hr />
<div>{{DISPLAYTITLE: Huge cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Critical points]]<br />
Huge cardinals (and their variants) were introduced by Kenneth Kunen in 1972 as a very large cardinal axiom. Kenneth Kunen first used them to prove that the consistency of the existence of a huge cardinal implies the consistency of $\text{ZFC}$+"there is a $\aleph_2$-[[filter|saturated ideal]] over $\omega_1$". <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
== Definitions ==<br />
<br />
Their formulation is similar to that of the formulation of [[superstrong]] cardinals. A huge cardinal is to a [[supercompact]] cardinal as a superstrong cardinal is to a [[strong]] cardinal, more precisely. The definition is part of a generalized phenomenon known as the "double helix", in which for some large cardinal properties $n$-$P_0$ and $n$-$P_1$, $n$-$P_0$ has less consistency strength than $n$-$P_1$, which has less consistency strength than $(n+1)$-$P_0$, and so on. This phenomenon is seen only around the [[n-fold variants|$n$-fold variants]] as of modern set theoretic concerns. <cite>Kentaro2007:DoubleHelix</cite><br />
<br />
Although they are very large, there is a first-order definition which is equivalent to $n$-hugeness, so the $\theta$-th $n$-huge cardinal is first-order definable whenever $\theta$ is first-order definable. This definition can be seen as a (very strong) strengthening of the first-order definition of [[measurable|measurability]].<br />
<br />
=== Elementary embedding definitions ===<br />
<br />
The elementary embedding definitions are somewhat standard. Let $j:V\rightarrow M$ be an [[elementary embedding]] with critical point $\kappa$ such that $M$ is a standard inner model of [[ZFC|$\text{ZFC}$]]. Then:<br />
<br />
*$\kappa$ is '''almost $n$-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length less than $\lambda$ (that is, $M^{<\lambda}\subset M$).<br />
*$\kappa$ is '''$n$-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length $\lambda$ ($M^\lambda\subset M$).<br />
*$\kappa$ is '''almost $n$-huge''' iff it is almost $n$-huge with target $\lambda$ for some $\lambda$.<br />
*$\kappa$ is '''$n$-huge''' iff it is $n$-huge with target $\lambda$ for some $\lambda$.<br />
*$\kappa$ is '''super almost $n$-huge''' iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is almost $n$-huge with target $\lambda$ (that is, the target can be made arbitrarily large).<br />
*$\kappa$ is '''super $n$-huge''' iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is $n$-huge with target $\lambda$.<br />
*$\kappa$ is '''almost huge''', '''huge''', '''super almost huge''', and '''superhuge''' iff it is '''almost $1$-huge''', '''$1$-huge''', etc. respectively.<br />
<br />
=== Ultrafilter definition ===<br />
<br />
The first-order definition of $n$-huge is somewhat similar to [[measurable|measurability]]. Specifically, $\kappa$ is measurable iff there is a nonprincipal $\kappa$-complete [[filter|ultrafilter]], $U$, over $\kappa$. A cardinal $\kappa$ is $n$-huge with target $\lambsa$ iff there is a nonprincipal $\kappa$-complete ultrafilter, $U$, over $\mathcal{P}(\lambda)$, and cardinals $\kappa=\lambda_0<\lambda_1<\lambda_2...<\lambda_{n-1}<\lambda_n=\lambda$ such that:<br />
<br />
$$\forall i<n\forall x\subseteq\lambda(ot(x\cap\lambda_{i+1})=\lambda_i\rightarrow x\in U)$$<br />
<br />
Where $ot(X)$ is the [[Order-isomorphism|order-type]] of the poset $(X,\in)$. <cite>Kanamori2009:HigherInfinite</cite> This definition is, more intuitively, making $U$ very large, like most ultrafilter characterizations of large cardinals ([[supercompact]], [[strongly compact]], etc.).<br />
$\kappa$ is then super $n$-huge if for all ordinals $\theta$ there is a $\lambda>\theta$ such that $\kappa$ is $n$-huge with target $\lambda$, i.e. $\lambda_n$ can be made arbitrarily large. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
== Consistency strength and size ==<br />
<br />
Hugeness exhibits a phenomenon associated with similarly defined large cardinals (the [[n-fold variants|$n$-fold variants]]) known as the ''double helix''. This phenomenon is when for one $n$-fold variant, letting a cardinal be called $n$-$P_0$ iff it has the property, and another variant, $n$-$P_1$, $n$-$P_0$ is weaker than $n$-$P_1$, which is weaker than $(n+1)$-$P_0$. <cite>Kentaro2007:DoubleHelix</cite> In the consistency strength hierarchy, here is where these lay (top being weakest):<br />
<br />
* [[measurable]] = $0$-[[superstrong]] = almost $0$-huge = super almost $0$-huge = $0$-huge = super $0$-huge <br />
* $n$-superstrong<br />
* $n$-fold supercompact<br />
* $(n+1)$-fold strong, $n$-fold extendible<br />
* $(n+1)$-fold Woodin, $n$-fold Vopěnka<br />
* $(n+1)$-fold Shelah<br />
* almost $n$-huge<br />
* super almost $n$-huge<br />
* $n$-huge<br />
* super $n$-huge<br />
* $(n+1)$-superstrong<br />
<br />
All huge variants lay at the top of the double helix restricted to some [[Omega|natural number]] $n$, although each are bested by [[rank-into-rank|I3]] cardinals (the [[elementary embedding|critical points]] of the I3 elementary embeddings). In fact, every I3 is preceeded by a stationary set of $n$-huge cardinals, for all $n$. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
Similarly, every huge cardinal $\kappa$ is almost huge, and there is a normal measure over $\kappa$ which contains every almost huge cardinal $\lambda<\kappa$. Every superhuge cardinal $\kappa$ is [[extendible]] and there is a normal measure over $\kappa$ which contains every extendible cardinal $\lambda<\kappa$. Every $(n+1)$-huge cardinal $\kappa$ has a normal measure which contains every cardinal $\lambda$ such that $V_\kappa\models$"$\lambda$ is super $n$-huge". <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
In terms of size, however, the least $n$-huge cardinal is smaller than the least [[supercompact]] cardinal. Assuming both exist, for any $\kappa$ which is supercompact and has an $n$-huge cardinal above it, there are $\kappa$ many $n$-huge cardinals less than $\kappa$. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
Every $n$-huge cardinal is $m$-huge for every $m\leq n$. Similarly with almost $n$-hugeness, super $n$-hugeness, and super almost $n$-hugeness. Every almost huge cardinal is [[Vopenka|Vopěnka]] (therefore the consistency of the existence of an almost-huge cardinal implies the consistency of Vopěnka's principle). <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Huge&diff=2091Huge2017-11-11T22:11:34Z<p>Wabb2t: /* Ultrafilter definition */</p>
<hr />
<div>{{DISPLAYTITLE: Huge cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Critical points]]<br />
Huge cardinals (and their variants) were introduced by Kenneth Kunen in 1972 as a very large cardinal axiom. Kenneth Kunen first used them to prove that the consistency of the existence of a huge cardinal implies the consistency of $\text{ZFC}$+"there is a $\aleph_2$-[[filter|saturated ideal]] over $\omega_1$". <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
== Definitions ==<br />
<br />
Their formulation is similar to that of the formulation of [[superstrong]] cardinals. A huge cardinal is to a [[supercompact]] cardinal as a superstrong cardinal is to a [[strong]] cardinal, more precisely. The definition is part of a generalized phenomenon known as the "double helix", in which for some large cardinal properties $n$-$P_0$ and $n$-$P_1$, $n$-$P_0$ has less consistency strength than $n$-$P_1$, which has less consistency strength than $(n+1)$-$P_0$, and so on. This phenomenon is seen only around the [[n-fold variants|$n$-fold variants]] as of modern set theoretic concerns. <cite>Kentaro2007:DoubleHelix</cite><br />
<br />
Although they are very large, there is a first-order definition which is equivalent to $n$-hugeness, so the $\theta$-th $n$-huge cardinal is first-order definable whenever $\theta$ is first-order definable. This definition can be seen as a (very strong) strengthening of the first-order definition of [[measurable|measurability]].<br />
<br />
=== Elementary embedding definitions ===<br />
<br />
The elementary embedding definitions are somewhat standard. Let $j:V\rightarrow M$ be an [[elementary embedding]] with critical point $\kappa$ such that $M$ is a standard inner model of [[ZFC|$\text{ZFC}$]]. Then:<br />
<br />
*$\kappa$ is '''almost $n$-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length less than $\lambda$ (that is, $M^{<\lambda}\subset M$).<br />
*$\kappa$ is '''$n$-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length $\lambda$ ($M^\lambda\subset M$).<br />
*$\kappa$ is '''almost $n$-huge''' iff it is almost $n$-huge with target $\lambda$ for some $\lambda$.<br />
*$\kappa$ is '''$n$-huge''' iff it is $n$-huge with target $\lambda$ for some $\lambda$.<br />
*$\kappa$ is '''super almost $n$-huge''' iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is almost $n$-huge with target $\lambda$ (that is, the target can be made arbitrarily large).<br />
*$\kappa$ is '''super $n$-huge''' iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is $n$-huge with target $\lambda$.<br />
*$\kappa$ is '''almost huge''', '''huge''', '''super almost huge''', and '''superhuge''' iff it is '''almost $1$-huge''', '''$1$-huge''', etc. respectively.<br />
<br />
=== Ultrafilter definition ===<br />
<br />
The first-order definition of $n$-huge is somewhat similar to [[measurable|measurability]]. Specifically, $\kappa$ is measurable iff there is a nonprincipal $\kappa$-complete [[filter|ultrafilter]], $U$, over $\kappa$. A cardinal $\kappa$ is $n$-huge with target $\lambsa$ iff there is a nonprincipal $\kappa$-complete ultrafilter, $U$, over $\mathcal{P}(\lambda)$, and cardinals $\kappa=\lambda_0<\lambda_1<\lambda_2...<\lambda_{n-1}<\lambda_n=\lambda$ such that:<br />
<br />
$$\forall i<n\forall x\subseteq\lambda(ot(x\cap\lambda_{i+1})=\lambda_i\rightarrow x\in U)$$<br />
<br />
Where $ot(X)$ is the [[Order-isomorphism|order-type]] of the poset $(X,\in)$. <cite>Kanamori2009:HigherInfinite</cite> This definition is, more intuitively, making $U$ very large, like most ultrafilter characterizations of large cardinals ([[supercompact]], [[strongly compact]], etc.).<br />
$\kappa$ is then super $n$-huge if for all ordinals $\theta$ there is a $\lambda>\theta$ such that $\kappa is $n$-huge with target $\lambda$, i.e. $\lambda_n$ can be made arbitrarily large. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
== Consistency strength and size ==<br />
<br />
Hugeness exhibits a phenomenon associated with similarly defined large cardinals (the [[n-fold variants|$n$-fold variants]]) known as the ''double helix''. This phenomenon is when for one $n$-fold variant, letting a cardinal be called $n$-$P_0$ iff it has the property, and another variant, $n$-$P_1$, $n$-$P_0$ is weaker than $n$-$P_1$, which is weaker than $(n+1)$-$P_0$. <cite>Kentaro2007:DoubleHelix</cite> In the consistency strength hierarchy, here is where these lay (top being weakest):<br />
<br />
* [[measurable]] = $0$-[[superstrong]] = almost $0$-huge = super almost $0$-huge = $0$-huge = super $0$-huge <br />
* $n$-superstrong<br />
* $n$-fold supercompact<br />
* $(n+1)$-fold strong, $n$-fold extendible<br />
* $(n+1)$-fold Woodin, $n$-fold Vopěnka<br />
* $(n+1)$-fold Shelah<br />
* almost $n$-huge<br />
* super almost $n$-huge<br />
* $n$-huge<br />
* super $n$-huge<br />
* $(n+1)$-superstrong<br />
<br />
All huge variants lay at the top of the double helix restricted to some [[Omega|natural number]] $n$, although each are bested by [[rank-into-rank|I3]] cardinals (the [[elementary embedding|critical points]] of the I3 elementary embeddings). In fact, every I3 is preceeded by a stationary set of $n$-huge cardinals, for all $n$. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
Similarly, every huge cardinal $\kappa$ is almost huge, and there is a normal measure over $\kappa$ which contains every almost huge cardinal $\lambda<\kappa$. Every superhuge cardinal $\kappa$ is [[extendible]] and there is a normal measure over $\kappa$ which contains every extendible cardinal $\lambda<\kappa$. Every $(n+1)$-huge cardinal $\kappa$ has a normal measure which contains every cardinal $\lambda$ such that $V_\kappa\models$"$\lambda$ is super $n$-huge". <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
In terms of size, however, the least $n$-huge cardinal is smaller than the least [[supercompact]] cardinal. Assuming both exist, for any $\kappa$ which is supercompact and has an $n$-huge cardinal above it, there are $\kappa$ many $n$-huge cardinals less than $\kappa$. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
Every $n$-huge cardinal is $m$-huge for every $m\leq n$. Similarly with almost $n$-hugeness, super $n$-hugeness, and super almost $n$-hugeness. Every almost huge cardinal is [[Vopenka|Vopěnka]] (therefore the consistency of the existence of an almost-huge cardinal implies the consistency of Vopěnka's principle). <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Filter&diff=2089Filter2017-11-11T21:57:49Z<p>Wabb2t: /* Measures */</p>
<hr />
<div>{{DISPLAYTITLE: Filter and ideals}}<br />
A ''filter'' on a set $S$ is a special subset of $\mathcal{P}(S)$ that contains $S$ itself, does not contain the [[empty set]], and is closed under finite intersections and the superset relation. An ''ideal'' on $S$ is the dual of a filter: if $F$ is a filter, the set of the complements (in $S$) of $F$'s elements forms an ideal, and vice-versa; equivalently, an ideal is a special subset of $\mathcal{P}(S)$ that contains the empty set but not $S$ itself, is closed under finite unions and the subset relation.<br />
<br />
An ''ultrafiler'' is a maximal filter, i.e. it is not a subset of any other filter, or equivalently, every subset of $S$ is either in it or its complement (in $S$) is. Filters, and especially ultrafilters, are closely connected to several large cardinal notions, such as [[measurable|measurable cardinals]] and [[strongly compact|strongly compact cardinals]]. The dual notion is a ''prime ideal''. Thus an ultrafilter and its dual prime ideal partitions $\mathcal{P}(S)$ in two.<br />
<br />
Intuitively, the members of a filter are the subsets of $S$ "large" enough to satisfy some property. $S$ is always "large enough", while $\empty$ never is. $F$ being closed under finite intersections means that the intersection of two large sets is still large enough - $F$'s sets only differ by a "too small" set. Also, $F$ being closed under the superset relation means that if a set $X$ contains a large enough set then $X$ is also large enough. For example, for any nonempty $X\subset S$, the set of all supersets of $X$ -i.e. the set of all sets "larger" than $X$ - is always a filter. Similarly, the members of an ideal will represent the subsets of $S$ "too small"; $\empty$ is always too small, $S$ never is, the union of two too small sets is still too small and if a set is contained (as a subset) in a too small set, then it is itself too small.<br />
<br />
== Definitions ==<br />
<br />
A set $F\subseteq\mathcal{P}(S)$ is a ''filter'' on $\mathcal{P}(S)$ (or just "on $S$") if it satisfies the following properties:<br />
* $\empty\not\in F$ (proper filter), $S\in F$<br />
* $X\cap Y\in F$ whenever $X,Y\in F$ (finite intersection property)<br />
* $Y\in F$ whenever $X\subset Y\subset S$ and $X\in F$ (upward closed / closed under supersets)<br />
<br />
A set $I\subseteq\mathcal{P}(S)$ is an ''ideal'' on $\mathcal{P}(S)$ (or just "on $S$") if it satisfies the following properties:<br />
* $S\not\in I$, $\empty\in I$<br />
* $X\cup Y\in F$ whenever $X,Y\in I$ (finite union property)<br />
* $Y\in I$ whenever $Y\subset X\subset S$ and $X\in I$ (downard closed / closed under subsets)<br />
<br />
Given a filter $F$, its ''dual ideal'' is $I=\{S\setminus X : X\in F\}$. Conversely, every ideal has a dual filter. If two filters/ideals are not equal, their duals aren't equal neither.<br />
<br />
A filter $F$ is ''trivial'' if $F=\{S\}$. It is ''principal'' if there exists $X\subset S$ such that $Y\in F$ if and only if $X\subset Y$. Every nonempty subset $X\subset S$ has an associated principal filter. Similarly, the trivial ideal is $I=\{\empty\}$, and an ideal is principal if there exists $X\subset S$ such that $Y\in I$ if and only if $Y\subset X$.<br />
<br />
A filter (resp. an ideal) $F$ is an ''ultrafilter'' (resp. a ''prime ideal'') if for all $X\subset S$, either $X\in F$ or $S\setminus X\in F$. Equivalently, there is no filter (resp. ideal) $F'$ such that $F\subset F'$ but $F\neq F'$ (i.e. $F$ is ''maximal'').<br />
<br />
$F$ is $\theta$-complete for a cardinal $\theta$ if for every family $\{X_\alpha : \alpha<\lambda\}$ with $\lambda<\theta$ and $X_\alpha\in F$ for all $\alpha<\lambda$, then $\bigcap_{\alpha<\lambda}X_\alpha\in F$. The completeness of $F$ is the smallest cardinal such that there is a subset $X\subset F$ such that $|X|=\theta$ and $\bigcap X\not\in F$, i.e. it is the largest $\theta$ such that $F$ is $\theta$-complete. Similarly for ideals, by replacing intersections by unions.<br />
<br />
A filter $F$ on $\kappa$ is ''normal'' if it is closed under diagonal intersections: $\Delta_{\alpha\in \kappa}X_\alpha = \{\xi\in \kappa : \xi\in\bigcap_{\alpha\in\xi}X_\alpha\}$. That is, for every family $\{X_\alpha : \alpha<\kappa\}$ and $X_\alpha\in F$ for all $\alpha<\kappa$, one have $\Delta_{\alpha<\kappa}X_\alpha\in F$. Similarly for ideals, by replacing intersections by unions.<br />
<br />
Whenever a filter is either nontrivial, nonprincipal, $\theta$-complete, normal or maximal, so is its dual ideal.<br />
<br />
== Properties ==<br />
<br />
The finite intersection property is equivalent to $\aleph_0$-completeness. Every set $G\subset \mathcal{P}(S)$ with the finite intersection property can be extended to a filter, i.e. there exists a filter $F$ such that $G\subset F$. A filter or an ideal being ''countably complete'' (or $\sigma$-complete) means that it is $\aleph_1$-complete. The completeness of a countably complete nonprincipal ultrafilter or prime ideal on S is always a [[measurable|measurable cardinal]]. However, every countably complete filter on a countable or finite set is principal.<br />
<br />
Every cardinal $\kappa\geq\aleph_0$ has $2^{2^\kappa}$ ultrafilters and prime ideals. Under the [[axiom of choice]], every filter can be extended to an ultrafilter, and every ideal can be extended to a prime ideal.<br />
<br />
If $G$ is a nonempty sets of filters on S, then $\bigcap G$ is a filter on S. If $G$ is a $\subset$-chain of filters, then $\bigcup G$ is a filter.<br />
<br />
Let $j:\mathcal{M}\to \mathcal{N}$ be a (nontrivial) [[elementary embedding]] with critical point $\kappa$. Then the set $\mathcal{U}_j=\{x\subset\kappa : \kappa\in j(x)\}$ is a $\kappa$-complete nonprincipal ultrafilter on $(\mathcal{P}(\kappa))^\mathcal{M}$; in particular if $\mathcal{M}=V$ then $(\mathcal{P}(\kappa))^\mathcal{M}=\mathcal{P}(\kappa)$ and thus $\kappa$ is [[measurable]].<br />
<br />
== The club filter, the non-stationary ideal and saturation ==<br />
<br />
Given a regular uncountable cardinal $\kappa$, the collection of all [[club|clubs]] in $\kappa$ has the finite intersection property, thus it can be extended to a filter. This filter contains precisely the subsets of $\kappa$ with a subset that is a club in $\kappa$. We we call this filter the ''club filter'' of $\kappa$. This filter is $\kappa$-complete and normal (i.e. closed under diagonal intersections).<br />
<br />
Let $I_{NS}$, the ''nonstationary ideal on $\kappa$'', be the dual ideal of the club filter of $\kappa$. This is a normal $\kappa$-complete ideal. Both $I_{NS}$ and the club filter are minimal: if $F$ is a normal filter containing all initial segements $\{\alpha : \alpha_0<\alpha<\kappa \}$ then it contains the club filter of $\kappa$. This means $I_{NS}$ and the clubfilter are not maximal, in particular the club filter is not a normal measure (see below) despite being normal and $\kappa$-complete.<br />
<br />
Let $I$ be a $\kappa$-complete ideal on $\kappa$ containing all singletons of elements of $\kappa$. $I$ contains all subsets of $\kappa$ of cardinality less than $\kappa$. We say that $I$ is $\lambda$-saturated if there is no collection $W$ of subsets of $\kappa$ such that $|W|=\lambda$, $I$ and $W$ are disjoint, but the intersection of any two elements of $W$ is in $I$. $\aleph_1$-saturation is called $\sigma$-saturation. $I$'s ''saturation'', $\text{sat}(I)$ is the smallest $\lambda$ such that $I$ is $\lambda$-saturated.<br />
<br />
An ideal $I$ is prime if and only if $\text{sat}(I)=2$. Trivially every ideal is $(2^\kappa)^+$-saturated. Any $\kappa$ carrying a $\sigma$-saturated $\kappa$-complete ideal must be either [[measurable]] or $\leq 2^{\aleph_0}$ and real-valued measurable. If there exists a $\kappa$-saturated $\kappa$-complete ideal on $\kappa$, then there is a such ideal that is additionally normal. Same for $\kappa^+$-saturation.<br />
<br />
If there exists a $\aleph_2$-saturated ideal on $\omega_1$ then:<br />
* $2^{\aleph_0}=\aleph_1$ implies $2^{\aleph_1}=\aleph_2$<br />
* $2^{\aleph_0}=\aleph_{\omega_1}$ implies $2^{\aleph_1}\leq\aleph_{\omega_2}$.<br />
* $\aleph_1 < 2^{\aleph_0} < \aleph_{\omega_1}$ implies $2^{\aleph_0} = 2^{\aleph_1}$.<br />
* $2^{<\aleph_{\omega_1}}=\aleph_{\omega_1}$ implies $2^{\aleph_{\omega_1}}<\aleph_{\omega_2}$<br />
This hypothesis, which follows from [[Martin's Maximum]], is consistent relative to a [[Woodin]] cardinal, in fact that ideal can be the nonstationary ideal on $\omega_1$. This cannot happen for cardinals larger than $\omega_1$ however: for every cardinal $\kappa\geq\aleph_2$, the nonstationary ideal on $\kappa$ is not $\kappa^+$-saturated.<br />
<br />
== Ultrapowers ==<br />
<br />
''Main article: [[Ultrapower]]''<br />
<br />
== Precipitous ideals ==<br />
<br />
''To be expanded.''<br />
<br />
== Measures ==<br />
<br />
Filters are related to the concept of ''measures''.<br />
<br />
Let $|S|\geq\aleph_0$. A (nontrivial $\sigma$-additive) ''measure'' on $S$ is a function $\mu:\mathcal{P}(S)\to[0,+\infty]$ such that:<br />
* $\mu(\empty)=0$, $\mu(S)>0$<br />
* $\mu(X)\leq\mu(Y)$ whenver $X\subset Y$<br />
* Let $\{X_n : n<\omega\}$ such that $X_i\cap X_j=\empty$ whenever $i<j$, then $\mu(\bigcup_{n<\omega}X_n)=\sum_{n=0}^{\infty}\mu(X_n)$<br />
<br />
$\mu$ is ''probabilist'' if $\mu(S)=1$. $\mu$ is ''nontrivial'' because there exists a set $A$ of positive measure, i.e. $\mu(A)>0$, since we required $\mu(S)>0$.<br />
<br />
$\mu$ is $\theta$-additive if $\{X_\alpha : \alpha<\lambda\}$ with $\lambda<\theta$ is such that $X_i\cap X_j=\empty$ whenever $i<j$, then $\mu(\bigcup_{\alpha<\lambda}X_\alpha)=\sum_{\alpha<\lambda}\mu(X_\alpha)$. Every measure $\mu$ is $\aleph_1$-additive (i.e. countably additive / $\sigma$-additive).<br />
<br />
$\mu$ is ''2-valued'' (or ''0-1-valued'') if for all $X\subset S$, either $\mu(X)=0$ or $\mu(X)=1$. A set $A\subset S$ such that $\mu(A)>0$ is an ''atom'' for $\mu$ if $\mu(X)=0$ or $\mu(X)=\mu(A)$ for all $X\subset A$. $\mu$ is ''atomless'' if it has no atoms.<br />
<br />
A set $X\subset S$ is ''null'' if $\mu(X)=0$.<br />
<br />
The [[:wikipedia:Lebesgue measure|Lebesgue measure]] is a certain kind of measure that is linked to the [[axiom of choice]] and to the [[axiom of determinacy]]. ''(See also [[projective]])''<br />
<br />
=== Properties ===<br />
<br />
* Let $\mu$ be a 2-valued measure on $S$. Then $\{X\subset S : \mu(X)=1\}$ is a $\sigma$-complete ultrafilter on $S$. Conversely, if $F$ is a $\sigma$-complete ultrafilter on $S$ then the funcion $\mu:P(S)\to[0,1]$ defined by "$\mu(X)=1$ if $X\in F$, $\mu(X)=0$ otherwise" is a 2-valued measure on $S$.<br />
<br />
* If $\mu$ has an atom $A$, the set $\{X\subset S : \mu(X\cap A)=\mu(A)\}$ is a $\sigma$-complete ultrafilter on $S$. <br />
<br />
* If $\mu$ is atomless (i.e. has no atoms), $\mu(\{x\})=0$ for every $x\in S$. In fact, $\mu(X)=0$ for every finite or countably infinite set $X\subset S$. Thus every measure on a countable set has an atom (otherwise $\mu(S)$ would be $0$, contradicting the nontriviality of $\mu$). <br />
<br />
* If $\mu$ is atomless, then every set $X\subset S$ of positive measure is the disjoint union of two sets of positive measure, also, $\mu$ has a continuum of different values and if $A$ is a set of positive measure, then for every $b\in [0;\mu(A)]$, there exists $B\subset A$ such that $\mu(B)=b$.<br />
<br />
== Normal fine measures and large cardinals ==<br />
<br />
Let $\kappa$ be an [[uncountable]] [[cardinal]]. Let $\mathcal{P}_\kappa(A)$ for $|A|\geq\kappa$ be the set of all subsets of $A$ of cardinality at most $\kappa$. <br />
<br />
A filter $F$ on $\mathcal{P}_\kappa(A)$ is a ''fine filter'' if for every $a\in A$, the set $\{x\in \mathcal{P}_\kappa(A) : a\in x\}\in F$. If $F$ is also $\sigma$-complete and an ultrafilter, it is called a ''fine measure'' because it can be identified with its dual measure $\mu$ defined by $\mu(X)=1$ iff $X\in F$, $0$ otherwise.<br />
<br />
A fine measure $F$ on $\mathcal{P}_\kappa(A)$ is ''normal'' if for every function $f:\mathcal{P}_\kappa(A)\to A$, if the set $\{x\in \mathcal{P}_\kappa(A) : f(x)\in x\}\in F$ then $f$ is constant on a set in $F$, i.e. there is $k\in A$ such that $F$ also contains the set $\{x\in \mathcal{P}_\kappa(A) : f(x)=k\}$. Note that normal fine measures are also normal in the sense that they are closed under diagonal intersections, i.e. for every family $\{X_\alpha : \alpha<\kappa\}$ and $X_\alpha\in F$ for all $\alpha<\kappa$, one has $\Delta_{\alpha<\kappa}X_\alpha\in F$.<br />
<br />
If there exists a 2-valued $\kappa$-additive measure on $\kappa$, then $\kappa$ is a [[measurable]] cardinal. This equivalent to saying that there is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$. If $j:V\to\mathcal{M}$ is a nontrivial elementary embedding with critical point $\kappa$, then $\mathcal{U}_j=\{x\subset\kappa : \kappa\in j(x)\}$ is a $\kappa$-complete nonprincipal ultrafilter on $\mathcal{P}(\kappa)$ and $\kappa$ is measurable. In fact, $\mathcal{U}_j$ is a normal fine measure on $\kappa$, which we can call the "canonical" normal fine measure generated by $j$.<br />
<br />
If, for every set $S$, every $\kappa$-complete filter on $S$ can be extended to a $\kappa$-complete ultrafilter on $S$, then $\kappa$ is '''[[strongly compact]]'''. The converse is also true, every strongly compact cardinal has this property. Not that nonprincipality is not required here. Every strongly compact cardinal is measurable, and it is consistent that the first measurable and the first strongly compact cardinals are equal. Strong compactness is furthermore equivalent to the assertion that for every set $A$ such that $|A|\geq\kappa$ there exists a fine measure on $\mathcal{P}_\kappa(A)$. Those measures don't have to be normal.<br />
<br />
If there is a set $\lambda$ with $\lambda\geq\kappa$ such that there is a normal fine measure on $\mathcal{P}_\kappa(\lambda)$, then $\kappa$ is $\lambda$-[[supercompact]]; if it is $\lambda$-supercompact for every $\lambda\geq\kappa$, then it is '''[[supercompact]]'''. This is equivalent to saying that for every set $A$ with $|A|\geq\kappa$, there is a normal fine measure on $\mathcal{P}_\kappa(A)$. Clearly, every supercompact is strongly compact by the last characterization of strong compactness. It is open whether supercompactness is stronger than strong compactness consistency-wise.<br />
<br />
Every set in a normal measure is [[stationary]], also every [[measurable]] cardinal carries a normal measure containing the set of all [[inaccessible]], [[Mahlo]], and even [[Ramsey]] cardinals below it. Every supercompact cardinal $\kappa$ carries $2^{2^\kappa}$ normal measures.<br />
<br />
== See Also ==<br />
<br />
* [[Filters on N|Filters on $\mathbb{N}$]]</div>Wabb2thttp://cantorsattic.info/index.php?title=Filter&diff=2088Filter2017-11-11T21:56:23Z<p>Wabb2t: /* The club filter, the non-stationary ideal and saturation */</p>
<hr />
<div>{{DISPLAYTITLE: Filter and ideals}}<br />
A ''filter'' on a set $S$ is a special subset of $\mathcal{P}(S)$ that contains $S$ itself, does not contain the [[empty set]], and is closed under finite intersections and the superset relation. An ''ideal'' on $S$ is the dual of a filter: if $F$ is a filter, the set of the complements (in $S$) of $F$'s elements forms an ideal, and vice-versa; equivalently, an ideal is a special subset of $\mathcal{P}(S)$ that contains the empty set but not $S$ itself, is closed under finite unions and the subset relation.<br />
<br />
An ''ultrafiler'' is a maximal filter, i.e. it is not a subset of any other filter, or equivalently, every subset of $S$ is either in it or its complement (in $S$) is. Filters, and especially ultrafilters, are closely connected to several large cardinal notions, such as [[measurable|measurable cardinals]] and [[strongly compact|strongly compact cardinals]]. The dual notion is a ''prime ideal''. Thus an ultrafilter and its dual prime ideal partitions $\mathcal{P}(S)$ in two.<br />
<br />
Intuitively, the members of a filter are the subsets of $S$ "large" enough to satisfy some property. $S$ is always "large enough", while $\empty$ never is. $F$ being closed under finite intersections means that the intersection of two large sets is still large enough - $F$'s sets only differ by a "too small" set. Also, $F$ being closed under the superset relation means that if a set $X$ contains a large enough set then $X$ is also large enough. For example, for any nonempty $X\subset S$, the set of all supersets of $X$ -i.e. the set of all sets "larger" than $X$ - is always a filter. Similarly, the members of an ideal will represent the subsets of $S$ "too small"; $\empty$ is always too small, $S$ never is, the union of two too small sets is still too small and if a set is contained (as a subset) in a too small set, then it is itself too small.<br />
<br />
== Definitions ==<br />
<br />
A set $F\subseteq\mathcal{P}(S)$ is a ''filter'' on $\mathcal{P}(S)$ (or just "on $S$") if it satisfies the following properties:<br />
* $\empty\not\in F$ (proper filter), $S\in F$<br />
* $X\cap Y\in F$ whenever $X,Y\in F$ (finite intersection property)<br />
* $Y\in F$ whenever $X\subset Y\subset S$ and $X\in F$ (upward closed / closed under supersets)<br />
<br />
A set $I\subseteq\mathcal{P}(S)$ is an ''ideal'' on $\mathcal{P}(S)$ (or just "on $S$") if it satisfies the following properties:<br />
* $S\not\in I$, $\empty\in I$<br />
* $X\cup Y\in F$ whenever $X,Y\in I$ (finite union property)<br />
* $Y\in I$ whenever $Y\subset X\subset S$ and $X\in I$ (downard closed / closed under subsets)<br />
<br />
Given a filter $F$, its ''dual ideal'' is $I=\{S\setminus X : X\in F\}$. Conversely, every ideal has a dual filter. If two filters/ideals are not equal, their duals aren't equal neither.<br />
<br />
A filter $F$ is ''trivial'' if $F=\{S\}$. It is ''principal'' if there exists $X\subset S$ such that $Y\in F$ if and only if $X\subset Y$. Every nonempty subset $X\subset S$ has an associated principal filter. Similarly, the trivial ideal is $I=\{\empty\}$, and an ideal is principal if there exists $X\subset S$ such that $Y\in I$ if and only if $Y\subset X$.<br />
<br />
A filter (resp. an ideal) $F$ is an ''ultrafilter'' (resp. a ''prime ideal'') if for all $X\subset S$, either $X\in F$ or $S\setminus X\in F$. Equivalently, there is no filter (resp. ideal) $F'$ such that $F\subset F'$ but $F\neq F'$ (i.e. $F$ is ''maximal'').<br />
<br />
$F$ is $\theta$-complete for a cardinal $\theta$ if for every family $\{X_\alpha : \alpha<\lambda\}$ with $\lambda<\theta$ and $X_\alpha\in F$ for all $\alpha<\lambda$, then $\bigcap_{\alpha<\lambda}X_\alpha\in F$. The completeness of $F$ is the smallest cardinal such that there is a subset $X\subset F$ such that $|X|=\theta$ and $\bigcap X\not\in F$, i.e. it is the largest $\theta$ such that $F$ is $\theta$-complete. Similarly for ideals, by replacing intersections by unions.<br />
<br />
A filter $F$ on $\kappa$ is ''normal'' if it is closed under diagonal intersections: $\Delta_{\alpha\in \kappa}X_\alpha = \{\xi\in \kappa : \xi\in\bigcap_{\alpha\in\xi}X_\alpha\}$. That is, for every family $\{X_\alpha : \alpha<\kappa\}$ and $X_\alpha\in F$ for all $\alpha<\kappa$, one have $\Delta_{\alpha<\kappa}X_\alpha\in F$. Similarly for ideals, by replacing intersections by unions.<br />
<br />
Whenever a filter is either nontrivial, nonprincipal, $\theta$-complete, normal or maximal, so is its dual ideal.<br />
<br />
== Properties ==<br />
<br />
The finite intersection property is equivalent to $\aleph_0$-completeness. Every set $G\subset \mathcal{P}(S)$ with the finite intersection property can be extended to a filter, i.e. there exists a filter $F$ such that $G\subset F$. A filter or an ideal being ''countably complete'' (or $\sigma$-complete) means that it is $\aleph_1$-complete. The completeness of a countably complete nonprincipal ultrafilter or prime ideal on S is always a [[measurable|measurable cardinal]]. However, every countably complete filter on a countable or finite set is principal.<br />
<br />
Every cardinal $\kappa\geq\aleph_0$ has $2^{2^\kappa}$ ultrafilters and prime ideals. Under the [[axiom of choice]], every filter can be extended to an ultrafilter, and every ideal can be extended to a prime ideal.<br />
<br />
If $G$ is a nonempty sets of filters on S, then $\bigcap G$ is a filter on S. If $G$ is a $\subset$-chain of filters, then $\bigcup G$ is a filter.<br />
<br />
Let $j:\mathcal{M}\to \mathcal{N}$ be a (nontrivial) [[elementary embedding]] with critical point $\kappa$. Then the set $\mathcal{U}_j=\{x\subset\kappa : \kappa\in j(x)\}$ is a $\kappa$-complete nonprincipal ultrafilter on $(\mathcal{P}(\kappa))^\mathcal{M}$; in particular if $\mathcal{M}=V$ then $(\mathcal{P}(\kappa))^\mathcal{M}=\mathcal{P}(\kappa)$ and thus $\kappa$ is [[measurable]].<br />
<br />
== The club filter, the non-stationary ideal and saturation ==<br />
<br />
Given a regular uncountable cardinal $\kappa$, the collection of all [[club|clubs]] in $\kappa$ has the finite intersection property, thus it can be extended to a filter. This filter contains precisely the subsets of $\kappa$ with a subset that is a club in $\kappa$. We we call this filter the ''club filter'' of $\kappa$. This filter is $\kappa$-complete and normal (i.e. closed under diagonal intersections).<br />
<br />
Let $I_{NS}$, the ''nonstationary ideal on $\kappa$'', be the dual ideal of the club filter of $\kappa$. This is a normal $\kappa$-complete ideal. Both $I_{NS}$ and the club filter are minimal: if $F$ is a normal filter containing all initial segements $\{\alpha : \alpha_0<\alpha<\kappa \}$ then it contains the club filter of $\kappa$. This means $I_{NS}$ and the clubfilter are not maximal, in particular the club filter is not a normal measure (see below) despite being normal and $\kappa$-complete.<br />
<br />
Let $I$ be a $\kappa$-complete ideal on $\kappa$ containing all singletons of elements of $\kappa$. $I$ contains all subsets of $\kappa$ of cardinality less than $\kappa$. We say that $I$ is $\lambda$-saturated if there is no collection $W$ of subsets of $\kappa$ such that $|W|=\lambda$, $I$ and $W$ are disjoint, but the intersection of any two elements of $W$ is in $I$. $\aleph_1$-saturation is called $\sigma$-saturation. $I$'s ''saturation'', $\text{sat}(I)$ is the smallest $\lambda$ such that $I$ is $\lambda$-saturated.<br />
<br />
An ideal $I$ is prime if and only if $\text{sat}(I)=2$. Trivially every ideal is $(2^\kappa)^+$-saturated. Any $\kappa$ carrying a $\sigma$-saturated $\kappa$-complete ideal must be either [[measurable]] or $\leq 2^{\aleph_0}$ and real-valued measurable. If there exists a $\kappa$-saturated $\kappa$-complete ideal on $\kappa$, then there is a such ideal that is additionally normal. Same for $\kappa^+$-saturation.<br />
<br />
If there exists a $\aleph_2$-saturated ideal on $\omega_1$ then:<br />
* $2^{\aleph_0}=\aleph_1$ implies $2^{\aleph_1}=\aleph_2$<br />
* $2^{\aleph_0}=\aleph_{\omega_1}$ implies $2^{\aleph_1}\leq\aleph_{\omega_2}$.<br />
* $\aleph_1 < 2^{\aleph_0} < \aleph_{\omega_1}$ implies $2^{\aleph_0} = 2^{\aleph_1}$.<br />
* $2^{<\aleph_{\omega_1}}=\aleph_{\omega_1}$ implies $2^{\aleph_{\omega_1}}<\aleph_{\omega_2}$<br />
This hypothesis, which follows from [[Martin's Maximum]], is consistent relative to a [[Woodin]] cardinal, in fact that ideal can be the nonstationary ideal on $\omega_1$. This cannot happen for cardinals larger than $\omega_1$ however: for every cardinal $\kappa\geq\aleph_2$, the nonstationary ideal on $\kappa$ is not $\kappa^+$-saturated.<br />
<br />
== Ultrapowers ==<br />
<br />
''Main article: [[Ultrapower]]''<br />
<br />
== Precipitous ideals ==<br />
<br />
''To be expanded.''<br />
<br />
== Measures ==<br />
<br />
Filters are related to the concept of ''measures''.<br />
<br />
Let $|S|\geq\aleph_0$. A (nontrivial $\sigma$-additive) ''measure'' on $S$ is a function $\mu:\mathcal{P}(S)\to[0,+\infty]$ such that:<br />
* $\mu(\empty)=0$, $\mu(S)>0$<br />
* $\mu(X)\leq\mu(Y)$ whenver $X\subset Y$<br />
* Let $\{X_n : n<\omega\}$ such that $X_i\cap X_j=\empty$ whenever $i<j$, then $\mu(\bigcup_{n<\omega}X_n)=\sum_{n=0}^{\infty}\mu(X_n)$<br />
<br />
$\mu$ is ''probabilist'' if $\mu(S)=1$. $\mu$ is ''nontrivial'' because there exists a set $A$ of positive measure, i.e. $\mu(A)>0$, since we required $\mu(S)>0$.<br />
<br />
$\mu$ is $\theta$-additive if $\{X_\alpha : \alpha<\lambda\}$ with $\lambda<\theta$ is such that $X_i\cap X_j=\empty$ whenever $i<j$, then $\mu(\bigcup_{\alpha<\lambda}X_\alpha)=\sum_{\alpha<\lambda}\mu(X_\alpha)$. Every measure $\mu$ is $\aleph_1$-additive (i.e. countably additive / $\sigma$-additive).<br />
<br />
$\mu$ is ''2-valued'' (or ''0-1-valued'') if for all $X\subset S$, either $\mu(X)=0$ or $\mu(X)=1$. A set $A\subset S$ such that $\mu(A)>0$ is an ''atom'' for $\mu$ if $\mu(X)=0$ or $\mu(X)=\mu(A)$ for all $X\subset A$. $\mu$ is ''atomless'' if it has no atoms.<br />
<br />
A set $X\subset S$ is ''null'' if $\mu(X)=0$.<br />
<br />
The [[:wikipedia:Lebesgue measure|Lebesgue measure]] is a certain kind of measure that is linked to the [[axiom of choice]] and to the [[axiom of determinacy]].<br />
<br />
=== Properties ===<br />
<br />
* Let $\mu$ be a 2-valued measure on $S$. Then $\{X\subset S : \mu(X)=1\}$ is a $\sigma$-complete ultrafilter on $S$. Conversely, if $F$ is a $\sigma$-complete ultrafilter on $S$ then the funcion $\mu:P(S)\to[0,1]$ defined by "$\mu(X)=1$ if $X\in F$, $\mu(X)=0$ otherwise" is a 2-valued measure on $S$.<br />
<br />
* If $\mu$ has an atom $A$, the set $\{X\subset S : \mu(X\cap A)=\mu(A)\}$ is a $\sigma$-complete ultrafilter on $S$. <br />
<br />
* If $\mu$ is atomless (i.e. has no atoms), $\mu(\{x\})=0$ for every $x\in S$. In fact, $\mu(X)=0$ for every finite or countably infinite set $X\subset S$. Thus every measure on a countable set has an atom (otherwise $\mu(S)$ would be $0$, contradicting the nontriviality of $\mu$). <br />
<br />
* If $\mu$ is atomless, then every set $X\subset S$ of positive measure is the disjoint union of two sets of positive measure, also, $\mu$ has a continuum of different values and if $A$ is a set of positive measure, then for every $b\in [0;\mu(A)]$, there exists $B\subset A$ such that $\mu(B)=b$.<br />
<br />
== Normal fine measures and large cardinals ==<br />
<br />
Let $\kappa$ be an [[uncountable]] [[cardinal]]. Let $\mathcal{P}_\kappa(A)$ for $|A|\geq\kappa$ be the set of all subsets of $A$ of cardinality at most $\kappa$. <br />
<br />
A filter $F$ on $\mathcal{P}_\kappa(A)$ is a ''fine filter'' if for every $a\in A$, the set $\{x\in \mathcal{P}_\kappa(A) : a\in x\}\in F$. If $F$ is also $\sigma$-complete and an ultrafilter, it is called a ''fine measure'' because it can be identified with its dual measure $\mu$ defined by $\mu(X)=1$ iff $X\in F$, $0$ otherwise.<br />
<br />
A fine measure $F$ on $\mathcal{P}_\kappa(A)$ is ''normal'' if for every function $f:\mathcal{P}_\kappa(A)\to A$, if the set $\{x\in \mathcal{P}_\kappa(A) : f(x)\in x\}\in F$ then $f$ is constant on a set in $F$, i.e. there is $k\in A$ such that $F$ also contains the set $\{x\in \mathcal{P}_\kappa(A) : f(x)=k\}$. Note that normal fine measures are also normal in the sense that they are closed under diagonal intersections, i.e. for every family $\{X_\alpha : \alpha<\kappa\}$ and $X_\alpha\in F$ for all $\alpha<\kappa$, one has $\Delta_{\alpha<\kappa}X_\alpha\in F$.<br />
<br />
If there exists a 2-valued $\kappa$-additive measure on $\kappa$, then $\kappa$ is a [[measurable]] cardinal. This equivalent to saying that there is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$. If $j:V\to\mathcal{M}$ is a nontrivial elementary embedding with critical point $\kappa$, then $\mathcal{U}_j=\{x\subset\kappa : \kappa\in j(x)\}$ is a $\kappa$-complete nonprincipal ultrafilter on $\mathcal{P}(\kappa)$ and $\kappa$ is measurable. In fact, $\mathcal{U}_j$ is a normal fine measure on $\kappa$, which we can call the "canonical" normal fine measure generated by $j$.<br />
<br />
If, for every set $S$, every $\kappa$-complete filter on $S$ can be extended to a $\kappa$-complete ultrafilter on $S$, then $\kappa$ is '''[[strongly compact]]'''. The converse is also true, every strongly compact cardinal has this property. Not that nonprincipality is not required here. Every strongly compact cardinal is measurable, and it is consistent that the first measurable and the first strongly compact cardinals are equal. Strong compactness is furthermore equivalent to the assertion that for every set $A$ such that $|A|\geq\kappa$ there exists a fine measure on $\mathcal{P}_\kappa(A)$. Those measures don't have to be normal.<br />
<br />
If there is a set $\lambda$ with $\lambda\geq\kappa$ such that there is a normal fine measure on $\mathcal{P}_\kappa(\lambda)$, then $\kappa$ is $\lambda$-[[supercompact]]; if it is $\lambda$-supercompact for every $\lambda\geq\kappa$, then it is '''[[supercompact]]'''. This is equivalent to saying that for every set $A$ with $|A|\geq\kappa$, there is a normal fine measure on $\mathcal{P}_\kappa(A)$. Clearly, every supercompact is strongly compact by the last characterization of strong compactness. It is open whether supercompactness is stronger than strong compactness consistency-wise.<br />
<br />
Every set in a normal measure is [[stationary]], also every [[measurable]] cardinal carries a normal measure containing the set of all [[inaccessible]], [[Mahlo]], and even [[Ramsey]] cardinals below it. Every supercompact cardinal $\kappa$ carries $2^{2^\kappa}$ normal measures.<br />
<br />
== See Also ==<br />
<br />
* [[Filters on N|Filters on $\mathbb{N}$]]</div>Wabb2thttp://cantorsattic.info/index.php?title=Filter&diff=2087Filter2017-11-11T21:52:01Z<p>Wabb2t: /* Normal fine measures and large cardinals */</p>
<hr />
<div>{{DISPLAYTITLE: Filter and ideals}}<br />
A ''filter'' on a set $S$ is a special subset of $\mathcal{P}(S)$ that contains $S$ itself, does not contain the [[empty set]], and is closed under finite intersections and the superset relation. An ''ideal'' on $S$ is the dual of a filter: if $F$ is a filter, the set of the complements (in $S$) of $F$'s elements forms an ideal, and vice-versa; equivalently, an ideal is a special subset of $\mathcal{P}(S)$ that contains the empty set but not $S$ itself, is closed under finite unions and the subset relation.<br />
<br />
An ''ultrafiler'' is a maximal filter, i.e. it is not a subset of any other filter, or equivalently, every subset of $S$ is either in it or its complement (in $S$) is. Filters, and especially ultrafilters, are closely connected to several large cardinal notions, such as [[measurable|measurable cardinals]] and [[strongly compact|strongly compact cardinals]]. The dual notion is a ''prime ideal''. Thus an ultrafilter and its dual prime ideal partitions $\mathcal{P}(S)$ in two.<br />
<br />
Intuitively, the members of a filter are the subsets of $S$ "large" enough to satisfy some property. $S$ is always "large enough", while $\empty$ never is. $F$ being closed under finite intersections means that the intersection of two large sets is still large enough - $F$'s sets only differ by a "too small" set. Also, $F$ being closed under the superset relation means that if a set $X$ contains a large enough set then $X$ is also large enough. For example, for any nonempty $X\subset S$, the set of all supersets of $X$ -i.e. the set of all sets "larger" than $X$ - is always a filter. Similarly, the members of an ideal will represent the subsets of $S$ "too small"; $\empty$ is always too small, $S$ never is, the union of two too small sets is still too small and if a set is contained (as a subset) in a too small set, then it is itself too small.<br />
<br />
== Definitions ==<br />
<br />
A set $F\subseteq\mathcal{P}(S)$ is a ''filter'' on $\mathcal{P}(S)$ (or just "on $S$") if it satisfies the following properties:<br />
* $\empty\not\in F$ (proper filter), $S\in F$<br />
* $X\cap Y\in F$ whenever $X,Y\in F$ (finite intersection property)<br />
* $Y\in F$ whenever $X\subset Y\subset S$ and $X\in F$ (upward closed / closed under supersets)<br />
<br />
A set $I\subseteq\mathcal{P}(S)$ is an ''ideal'' on $\mathcal{P}(S)$ (or just "on $S$") if it satisfies the following properties:<br />
* $S\not\in I$, $\empty\in I$<br />
* $X\cup Y\in F$ whenever $X,Y\in I$ (finite union property)<br />
* $Y\in I$ whenever $Y\subset X\subset S$ and $X\in I$ (downard closed / closed under subsets)<br />
<br />
Given a filter $F$, its ''dual ideal'' is $I=\{S\setminus X : X\in F\}$. Conversely, every ideal has a dual filter. If two filters/ideals are not equal, their duals aren't equal neither.<br />
<br />
A filter $F$ is ''trivial'' if $F=\{S\}$. It is ''principal'' if there exists $X\subset S$ such that $Y\in F$ if and only if $X\subset Y$. Every nonempty subset $X\subset S$ has an associated principal filter. Similarly, the trivial ideal is $I=\{\empty\}$, and an ideal is principal if there exists $X\subset S$ such that $Y\in I$ if and only if $Y\subset X$.<br />
<br />
A filter (resp. an ideal) $F$ is an ''ultrafilter'' (resp. a ''prime ideal'') if for all $X\subset S$, either $X\in F$ or $S\setminus X\in F$. Equivalently, there is no filter (resp. ideal) $F'$ such that $F\subset F'$ but $F\neq F'$ (i.e. $F$ is ''maximal'').<br />
<br />
$F$ is $\theta$-complete for a cardinal $\theta$ if for every family $\{X_\alpha : \alpha<\lambda\}$ with $\lambda<\theta$ and $X_\alpha\in F$ for all $\alpha<\lambda$, then $\bigcap_{\alpha<\lambda}X_\alpha\in F$. The completeness of $F$ is the smallest cardinal such that there is a subset $X\subset F$ such that $|X|=\theta$ and $\bigcap X\not\in F$, i.e. it is the largest $\theta$ such that $F$ is $\theta$-complete. Similarly for ideals, by replacing intersections by unions.<br />
<br />
A filter $F$ on $\kappa$ is ''normal'' if it is closed under diagonal intersections: $\Delta_{\alpha\in \kappa}X_\alpha = \{\xi\in \kappa : \xi\in\bigcap_{\alpha\in\xi}X_\alpha\}$. That is, for every family $\{X_\alpha : \alpha<\kappa\}$ and $X_\alpha\in F$ for all $\alpha<\kappa$, one have $\Delta_{\alpha<\kappa}X_\alpha\in F$. Similarly for ideals, by replacing intersections by unions.<br />
<br />
Whenever a filter is either nontrivial, nonprincipal, $\theta$-complete, normal or maximal, so is its dual ideal.<br />
<br />
== Properties ==<br />
<br />
The finite intersection property is equivalent to $\aleph_0$-completeness. Every set $G\subset \mathcal{P}(S)$ with the finite intersection property can be extended to a filter, i.e. there exists a filter $F$ such that $G\subset F$. A filter or an ideal being ''countably complete'' (or $\sigma$-complete) means that it is $\aleph_1$-complete. The completeness of a countably complete nonprincipal ultrafilter or prime ideal on S is always a [[measurable|measurable cardinal]]. However, every countably complete filter on a countable or finite set is principal.<br />
<br />
Every cardinal $\kappa\geq\aleph_0$ has $2^{2^\kappa}$ ultrafilters and prime ideals. Under the [[axiom of choice]], every filter can be extended to an ultrafilter, and every ideal can be extended to a prime ideal.<br />
<br />
If $G$ is a nonempty sets of filters on S, then $\bigcap G$ is a filter on S. If $G$ is a $\subset$-chain of filters, then $\bigcup G$ is a filter.<br />
<br />
Let $j:\mathcal{M}\to \mathcal{N}$ be a (nontrivial) [[elementary embedding]] with critical point $\kappa$. Then the set $\mathcal{U}_j=\{x\subset\kappa : \kappa\in j(x)\}$ is a $\kappa$-complete nonprincipal ultrafilter on $(\mathcal{P}(\kappa))^\mathcal{M}$; in particular if $\mathcal{M}=V$ then $(\mathcal{P}(\kappa))^\mathcal{M}=\mathcal{P}(\kappa)$ and thus $\kappa$ is [[measurable]].<br />
<br />
== The club filter, the non-stationary ideal and saturation ==<br />
<br />
Given a regular uncountable cardinal $\kappa$, the collection of all [[club|clubs]] in $\kappa$ has the finite intersection property, thus it can be extended to a filter. This filter contains precisely the subsets of $\kappa$ with a subset that is a club in $\kappa$. We we call this filter the ''club filter'' of $\kappa$. This filter is $\kappa$-complete and normal (i.e. closed under diagonal intersections).<br />
<br />
Let $I_{NS}$, the ''nonstationary ideal on $\kappa$'', be the dual ideal of the club filter of $\kappa$. This is a normal $\kappa$-complete ideal. Both $I_{NS}$ and the club filter are minimal: if $F$ is a normal filter containing all initial segements $\{\alpha : \alpha_0<\alpha<\kappa \}$ then it contains the club filter of $\kappa$. This means $I_{NS}$ and the clubfilter are not maximal, in particular the club filter is not a normal measure (see below) despite being normal and $\kappa$-complete.<br />
<br />
Let $I$ be a $\kappa$-complete ideal on $\kappa$ containing all singletons of elements of $\kappa$. $I$ contains all subsets of $\kappa$ of cardinality less than $\kappa$. We say that $I$ is $\lambda$-saturated if there is no collection $W$ of subsets of $\kappa$ such that $|W|=\lambda$, $I$ and $W$ are disjoint, but the intersection of any two elements of $W$ is in $I$. $\aleph_1$-saturation is called $\sigma$-saturation. $sat(I)$ is the smallest $\lambda$ such that $I$ is $\lambda$-saturated.<br />
<br />
An ideal $I$ is prime if and only if $sat(I)=2$. Trivially every ideal is $(2^\kappa)^+$-saturated. Any $\kappa$ carrying a $\sigma$-saturated $\kappa$-complete ideal must be either [[measurable]] or $\leq 2^{\aleph_0}$ and real-valued measurable. If there exists a $\kappa$-saturated $\kappa$-complete ideal on $\kappa$, then there is a such ideal that is additionally normal. Same for $\kappa^+$-saturation.<br />
<br />
If there exists a $\aleph_2$-saturated ideal on $\omega_1$ then:<br />
* $2^{\aleph_0}=\aleph_1$ implies $2^{\aleph_1}=\aleph_2$<br />
* $2^{\aleph_0}=\aleph_{\omega_1}$ implies $2^{\aleph_1}\leq\aleph_{\omega_2}$.<br />
* $\aleph_1 < 2^{\aleph_0} < \aleph_{\omega_1}$ implies $2^{\aleph_0} = 2^{\aleph_1}$.<br />
* $2^{<\aleph_{\omega_1}}=\aleph_{\omega_1}$ implies $2^{\aleph_{\omega_1}}<\aleph_{\omega_2}$<br />
This hypothesis, which follows from [[Martin's Maximum]], is consistent relative to a [[Woodin]] cardinal, in fact that ideal can be the nonstationary ideal on $\omega_1$. This cannot happen for cardinals larger than $\omega_1$ however: for every cardinal $\kappa\geq\aleph_2$, the nonstationary ideal on $\kappa$ is not $\kappa^+$-saturated.<br />
<br />
== Ultrapowers ==<br />
<br />
''Main article: [[Ultrapower]]''<br />
<br />
== Precipitous ideals ==<br />
<br />
''To be expanded.''<br />
<br />
== Measures ==<br />
<br />
Filters are related to the concept of ''measures''.<br />
<br />
Let $|S|\geq\aleph_0$. A (nontrivial $\sigma$-additive) ''measure'' on $S$ is a function $\mu:\mathcal{P}(S)\to[0,+\infty]$ such that:<br />
* $\mu(\empty)=0$, $\mu(S)>0$<br />
* $\mu(X)\leq\mu(Y)$ whenver $X\subset Y$<br />
* Let $\{X_n : n<\omega\}$ such that $X_i\cap X_j=\empty$ whenever $i<j$, then $\mu(\bigcup_{n<\omega}X_n)=\sum_{n=0}^{\infty}\mu(X_n)$<br />
<br />
$\mu$ is ''probabilist'' if $\mu(S)=1$. $\mu$ is ''nontrivial'' because there exists a set $A$ of positive measure, i.e. $\mu(A)>0$, since we required $\mu(S)>0$.<br />
<br />
$\mu$ is $\theta$-additive if $\{X_\alpha : \alpha<\lambda\}$ with $\lambda<\theta$ is such that $X_i\cap X_j=\empty$ whenever $i<j$, then $\mu(\bigcup_{\alpha<\lambda}X_\alpha)=\sum_{\alpha<\lambda}\mu(X_\alpha)$. Every measure $\mu$ is $\aleph_1$-additive (i.e. countably additive / $\sigma$-additive).<br />
<br />
$\mu$ is ''2-valued'' (or ''0-1-valued'') if for all $X\subset S$, either $\mu(X)=0$ or $\mu(X)=1$. A set $A\subset S$ such that $\mu(A)>0$ is an ''atom'' for $\mu$ if $\mu(X)=0$ or $\mu(X)=\mu(A)$ for all $X\subset A$. $\mu$ is ''atomless'' if it has no atoms.<br />
<br />
A set $X\subset S$ is ''null'' if $\mu(X)=0$.<br />
<br />
The [[:wikipedia:Lebesgue measure|Lebesgue measure]] is a certain kind of measure that is linked to the [[axiom of choice]] and to the [[axiom of determinacy]].<br />
<br />
=== Properties ===<br />
<br />
* Let $\mu$ be a 2-valued measure on $S$. Then $\{X\subset S : \mu(X)=1\}$ is a $\sigma$-complete ultrafilter on $S$. Conversely, if $F$ is a $\sigma$-complete ultrafilter on $S$ then the funcion $\mu:P(S)\to[0,1]$ defined by "$\mu(X)=1$ if $X\in F$, $\mu(X)=0$ otherwise" is a 2-valued measure on $S$.<br />
<br />
* If $\mu$ has an atom $A$, the set $\{X\subset S : \mu(X\cap A)=\mu(A)\}$ is a $\sigma$-complete ultrafilter on $S$. <br />
<br />
* If $\mu$ is atomless (i.e. has no atoms), $\mu(\{x\})=0$ for every $x\in S$. In fact, $\mu(X)=0$ for every finite or countably infinite set $X\subset S$. Thus every measure on a countable set has an atom (otherwise $\mu(S)$ would be $0$, contradicting the nontriviality of $\mu$). <br />
<br />
* If $\mu$ is atomless, then every set $X\subset S$ of positive measure is the disjoint union of two sets of positive measure, also, $\mu$ has a continuum of different values and if $A$ is a set of positive measure, then for every $b\in [0;\mu(A)]$, there exists $B\subset A$ such that $\mu(B)=b$.<br />
<br />
== Normal fine measures and large cardinals ==<br />
<br />
Let $\kappa$ be an [[uncountable]] [[cardinal]]. Let $\mathcal{P}_\kappa(A)$ for $|A|\geq\kappa$ be the set of all subsets of $A$ of cardinality at most $\kappa$. <br />
<br />
A filter $F$ on $\mathcal{P}_\kappa(A)$ is a ''fine filter'' if for every $a\in A$, the set $\{x\in \mathcal{P}_\kappa(A) : a\in x\}\in F$. If $F$ is also $\sigma$-complete and an ultrafilter, it is called a ''fine measure'' because it can be identified with its dual measure $\mu$ defined by $\mu(X)=1$ iff $X\in F$, $0$ otherwise.<br />
<br />
A fine measure $F$ on $\mathcal{P}_\kappa(A)$ is ''normal'' if for every function $f:\mathcal{P}_\kappa(A)\to A$, if the set $\{x\in \mathcal{P}_\kappa(A) : f(x)\in x\}\in F$ then $f$ is constant on a set in $F$, i.e. there is $k\in A$ such that $F$ also contains the set $\{x\in \mathcal{P}_\kappa(A) : f(x)=k\}$. Note that normal fine measures are also normal in the sense that they are closed under diagonal intersections, i.e. for every family $\{X_\alpha : \alpha<\kappa\}$ and $X_\alpha\in F$ for all $\alpha<\kappa$, one has $\Delta_{\alpha<\kappa}X_\alpha\in F$.<br />
<br />
If there exists a 2-valued $\kappa$-additive measure on $\kappa$, then $\kappa$ is a [[measurable]] cardinal. This equivalent to saying that there is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$. If $j:V\to\mathcal{M}$ is a nontrivial elementary embedding with critical point $\kappa$, then $\mathcal{U}_j=\{x\subset\kappa : \kappa\in j(x)\}$ is a $\kappa$-complete nonprincipal ultrafilter on $\mathcal{P}(\kappa)$ and $\kappa$ is measurable. In fact, $\mathcal{U}_j$ is a normal fine measure on $\kappa$, which we can call the "canonical" normal fine measure generated by $j$.<br />
<br />
If, for every set $S$, every $\kappa$-complete filter on $S$ can be extended to a $\kappa$-complete ultrafilter on $S$, then $\kappa$ is '''[[strongly compact]]'''. The converse is also true, every strongly compact cardinal has this property. Not that nonprincipality is not required here. Every strongly compact cardinal is measurable, and it is consistent that the first measurable and the first strongly compact cardinals are equal. Strong compactness is furthermore equivalent to the assertion that for every set $A$ such that $|A|\geq\kappa$ there exists a fine measure on $\mathcal{P}_\kappa(A)$. Those measures don't have to be normal.<br />
<br />
If there is a set $\lambda$ with $\lambda\geq\kappa$ such that there is a normal fine measure on $\mathcal{P}_\kappa(\lambda)$, then $\kappa$ is $\lambda$-[[supercompact]]; if it is $\lambda$-supercompact for every $\lambda\geq\kappa$, then it is '''[[supercompact]]'''. This is equivalent to saying that for every set $A$ with $|A|\geq\kappa$, there is a normal fine measure on $\mathcal{P}_\kappa(A)$. Clearly, every supercompact is strongly compact by the last characterization of strong compactness. It is open whether supercompactness is stronger than strong compactness consistency-wise.<br />
<br />
Every set in a normal measure is [[stationary]], also every [[measurable]] cardinal carries a normal measure containing the set of all [[inaccessible]], [[Mahlo]], and even [[Ramsey]] cardinals below it. Every supercompact cardinal $\kappa$ carries $2^{2^\kappa}$ normal measures.<br />
<br />
== See Also ==<br />
<br />
* [[Filters on N|Filters on $\mathbb{N}$]]</div>Wabb2thttp://cantorsattic.info/index.php?title=Filter&diff=2086Filter2017-11-11T21:43:24Z<p>Wabb2t: /* Normal fine measures and large cardinals */</p>
<hr />
<div>{{DISPLAYTITLE: Filter and ideals}}<br />
A ''filter'' on a set $S$ is a special subset of $\mathcal{P}(S)$ that contains $S$ itself, does not contain the [[empty set]], and is closed under finite intersections and the superset relation. An ''ideal'' on $S$ is the dual of a filter: if $F$ is a filter, the set of the complements (in $S$) of $F$'s elements forms an ideal, and vice-versa; equivalently, an ideal is a special subset of $\mathcal{P}(S)$ that contains the empty set but not $S$ itself, is closed under finite unions and the subset relation.<br />
<br />
An ''ultrafiler'' is a maximal filter, i.e. it is not a subset of any other filter, or equivalently, every subset of $S$ is either in it or its complement (in $S$) is. Filters, and especially ultrafilters, are closely connected to several large cardinal notions, such as [[measurable|measurable cardinals]] and [[strongly compact|strongly compact cardinals]]. The dual notion is a ''prime ideal''. Thus an ultrafilter and its dual prime ideal partitions $\mathcal{P}(S)$ in two.<br />
<br />
Intuitively, the members of a filter are the subsets of $S$ "large" enough to satisfy some property. $S$ is always "large enough", while $\empty$ never is. $F$ being closed under finite intersections means that the intersection of two large sets is still large enough - $F$'s sets only differ by a "too small" set. Also, $F$ being closed under the superset relation means that if a set $X$ contains a large enough set then $X$ is also large enough. For example, for any nonempty $X\subset S$, the set of all supersets of $X$ -i.e. the set of all sets "larger" than $X$ - is always a filter. Similarly, the members of an ideal will represent the subsets of $S$ "too small"; $\empty$ is always too small, $S$ never is, the union of two too small sets is still too small and if a set is contained (as a subset) in a too small set, then it is itself too small.<br />
<br />
== Definitions ==<br />
<br />
A set $F\subseteq\mathcal{P}(S)$ is a ''filter'' on $\mathcal{P}(S)$ (or just "on $S$") if it satisfies the following properties:<br />
* $\empty\not\in F$ (proper filter), $S\in F$<br />
* $X\cap Y\in F$ whenever $X,Y\in F$ (finite intersection property)<br />
* $Y\in F$ whenever $X\subset Y\subset S$ and $X\in F$ (upward closed / closed under supersets)<br />
<br />
A set $I\subseteq\mathcal{P}(S)$ is an ''ideal'' on $\mathcal{P}(S)$ (or just "on $S$") if it satisfies the following properties:<br />
* $S\not\in I$, $\empty\in I$<br />
* $X\cup Y\in F$ whenever $X,Y\in I$ (finite union property)<br />
* $Y\in I$ whenever $Y\subset X\subset S$ and $X\in I$ (downard closed / closed under subsets)<br />
<br />
Given a filter $F$, its ''dual ideal'' is $I=\{S\setminus X : X\in F\}$. Conversely, every ideal has a dual filter. If two filters/ideals are not equal, their duals aren't equal neither.<br />
<br />
A filter $F$ is ''trivial'' if $F=\{S\}$. It is ''principal'' if there exists $X\subset S$ such that $Y\in F$ if and only if $X\subset Y$. Every nonempty subset $X\subset S$ has an associated principal filter. Similarly, the trivial ideal is $I=\{\empty\}$, and an ideal is principal if there exists $X\subset S$ such that $Y\in I$ if and only if $Y\subset X$.<br />
<br />
A filter (resp. an ideal) $F$ is an ''ultrafilter'' (resp. a ''prime ideal'') if for all $X\subset S$, either $X\in F$ or $S\setminus X\in F$. Equivalently, there is no filter (resp. ideal) $F'$ such that $F\subset F'$ but $F\neq F'$ (i.e. $F$ is ''maximal'').<br />
<br />
$F$ is $\theta$-complete for a cardinal $\theta$ if for every family $\{X_\alpha : \alpha<\lambda\}$ with $\lambda<\theta$ and $X_\alpha\in F$ for all $\alpha<\lambda$, then $\bigcap_{\alpha<\lambda}X_\alpha\in F$. The completeness of $F$ is the smallest cardinal such that there is a subset $X\subset F$ such that $|X|=\theta$ and $\bigcap X\not\in F$, i.e. it is the largest $\theta$ such that $F$ is $\theta$-complete. Similarly for ideals, by replacing intersections by unions.<br />
<br />
A filter $F$ on $\kappa$ is ''normal'' if it is closed under diagonal intersections: $\Delta_{\alpha\in \kappa}X_\alpha = \{\xi\in \kappa : \xi\in\bigcap_{\alpha\in\xi}X_\alpha\}$. That is, for every family $\{X_\alpha : \alpha<\kappa\}$ and $X_\alpha\in F$ for all $\alpha<\kappa$, one have $\Delta_{\alpha<\kappa}X_\alpha\in F$. Similarly for ideals, by replacing intersections by unions.<br />
<br />
Whenever a filter is either nontrivial, nonprincipal, $\theta$-complete, normal or maximal, so is its dual ideal.<br />
<br />
== Properties ==<br />
<br />
The finite intersection property is equivalent to $\aleph_0$-completeness. Every set $G\subset \mathcal{P}(S)$ with the finite intersection property can be extended to a filter, i.e. there exists a filter $F$ such that $G\subset F$. A filter or an ideal being ''countably complete'' (or $\sigma$-complete) means that it is $\aleph_1$-complete. The completeness of a countably complete nonprincipal ultrafilter or prime ideal on S is always a [[measurable|measurable cardinal]]. However, every countably complete filter on a countable or finite set is principal.<br />
<br />
Every cardinal $\kappa\geq\aleph_0$ has $2^{2^\kappa}$ ultrafilters and prime ideals. Under the [[axiom of choice]], every filter can be extended to an ultrafilter, and every ideal can be extended to a prime ideal.<br />
<br />
If $G$ is a nonempty sets of filters on S, then $\bigcap G$ is a filter on S. If $G$ is a $\subset$-chain of filters, then $\bigcup G$ is a filter.<br />
<br />
Let $j:\mathcal{M}\to \mathcal{N}$ be a (nontrivial) [[elementary embedding]] with critical point $\kappa$. Then the set $\mathcal{U}_j=\{x\subset\kappa : \kappa\in j(x)\}$ is a $\kappa$-complete nonprincipal ultrafilter on $(\mathcal{P}(\kappa))^\mathcal{M}$; in particular if $\mathcal{M}=V$ then $(\mathcal{P}(\kappa))^\mathcal{M}=\mathcal{P}(\kappa)$ and thus $\kappa$ is [[measurable]].<br />
<br />
== The club filter, the non-stationary ideal and saturation ==<br />
<br />
Given a regular uncountable cardinal $\kappa$, the collection of all [[club|clubs]] in $\kappa$ has the finite intersection property, thus it can be extended to a filter. This filter contains precisely the subsets of $\kappa$ with a subset that is a club in $\kappa$. We we call this filter the ''club filter'' of $\kappa$. This filter is $\kappa$-complete and normal (i.e. closed under diagonal intersections).<br />
<br />
Let $I_{NS}$, the ''nonstationary ideal on $\kappa$'', be the dual ideal of the club filter of $\kappa$. This is a normal $\kappa$-complete ideal. Both $I_{NS}$ and the club filter are minimal: if $F$ is a normal filter containing all initial segements $\{\alpha : \alpha_0<\alpha<\kappa \}$ then it contains the club filter of $\kappa$. This means $I_{NS}$ and the clubfilter are not maximal, in particular the club filter is not a normal measure (see below) despite being normal and $\kappa$-complete.<br />
<br />
Let $I$ be a $\kappa$-complete ideal on $\kappa$ containing all singletons of elements of $\kappa$. $I$ contains all subsets of $\kappa$ of cardinality less than $\kappa$. We say that $I$ is $\lambda$-saturated if there is no collection $W$ of subsets of $\kappa$ such that $|W|=\lambda$, $I$ and $W$ are disjoint, but the intersection of any two elements of $W$ is in $I$. $\aleph_1$-saturation is called $\sigma$-saturation. $sat(I)$ is the smallest $\lambda$ such that $I$ is $\lambda$-saturated.<br />
<br />
An ideal $I$ is prime if and only if $sat(I)=2$. Trivially every ideal is $(2^\kappa)^+$-saturated. Any $\kappa$ carrying a $\sigma$-saturated $\kappa$-complete ideal must be either [[measurable]] or $\leq 2^{\aleph_0}$ and real-valued measurable. If there exists a $\kappa$-saturated $\kappa$-complete ideal on $\kappa$, then there is a such ideal that is additionally normal. Same for $\kappa^+$-saturation.<br />
<br />
If there exists a $\aleph_2$-saturated ideal on $\omega_1$ then:<br />
* $2^{\aleph_0}=\aleph_1$ implies $2^{\aleph_1}=\aleph_2$<br />
* $2^{\aleph_0}=\aleph_{\omega_1}$ implies $2^{\aleph_1}\leq\aleph_{\omega_2}$.<br />
* $\aleph_1 < 2^{\aleph_0} < \aleph_{\omega_1}$ implies $2^{\aleph_0} = 2^{\aleph_1}$.<br />
* $2^{<\aleph_{\omega_1}}=\aleph_{\omega_1}$ implies $2^{\aleph_{\omega_1}}<\aleph_{\omega_2}$<br />
This hypothesis, which follows from [[Martin's Maximum]], is consistent relative to a [[Woodin]] cardinal, in fact that ideal can be the nonstationary ideal on $\omega_1$. This cannot happen for cardinals larger than $\omega_1$ however: for every cardinal $\kappa\geq\aleph_2$, the nonstationary ideal on $\kappa$ is not $\kappa^+$-saturated.<br />
<br />
== Ultrapowers ==<br />
<br />
''Main article: [[Ultrapower]]''<br />
<br />
== Precipitous ideals ==<br />
<br />
''To be expanded.''<br />
<br />
== Measures ==<br />
<br />
Filters are related to the concept of ''measures''.<br />
<br />
Let $|S|\geq\aleph_0$. A (nontrivial $\sigma$-additive) ''measure'' on $S$ is a function $\mu:\mathcal{P}(S)\to[0,+\infty]$ such that:<br />
* $\mu(\empty)=0$, $\mu(S)>0$<br />
* $\mu(X)\leq\mu(Y)$ whenver $X\subset Y$<br />
* Let $\{X_n : n<\omega\}$ such that $X_i\cap X_j=\empty$ whenever $i<j$, then $\mu(\bigcup_{n<\omega}X_n)=\sum_{n=0}^{\infty}\mu(X_n)$<br />
<br />
$\mu$ is ''probabilist'' if $\mu(S)=1$. $\mu$ is ''nontrivial'' because there exists a set $A$ of positive measure, i.e. $\mu(A)>0$, since we required $\mu(S)>0$.<br />
<br />
$\mu$ is $\theta$-additive if $\{X_\alpha : \alpha<\lambda\}$ with $\lambda<\theta$ is such that $X_i\cap X_j=\empty$ whenever $i<j$, then $\mu(\bigcup_{\alpha<\lambda}X_\alpha)=\sum_{\alpha<\lambda}\mu(X_\alpha)$. Every measure $\mu$ is $\aleph_1$-additive (i.e. countably additive / $\sigma$-additive).<br />
<br />
$\mu$ is ''2-valued'' (or ''0-1-valued'') if for all $X\subset S$, either $\mu(X)=0$ or $\mu(X)=1$. A set $A\subset S$ such that $\mu(A)>0$ is an ''atom'' for $\mu$ if $\mu(X)=0$ or $\mu(X)=\mu(A)$ for all $X\subset A$. $\mu$ is ''atomless'' if it has no atoms.<br />
<br />
A set $X\subset S$ is ''null'' if $\mu(X)=0$.<br />
<br />
The [[:wikipedia:Lebesgue measure|Lebesgue measure]] is a certain kind of measure that is linked to the [[axiom of choice]] and to the [[axiom of determinacy]].<br />
<br />
=== Properties ===<br />
<br />
* Let $\mu$ be a 2-valued measure on $S$. Then $\{X\subset S : \mu(X)=1\}$ is a $\sigma$-complete ultrafilter on $S$. Conversely, if $F$ is a $\sigma$-complete ultrafilter on $S$ then the funcion $\mu:P(S)\to[0,1]$ defined by "$\mu(X)=1$ if $X\in F$, $\mu(X)=0$ otherwise" is a 2-valued measure on $S$.<br />
<br />
* If $\mu$ has an atom $A$, the set $\{X\subset S : \mu(X\cap A)=\mu(A)\}$ is a $\sigma$-complete ultrafilter on $S$. <br />
<br />
* If $\mu$ is atomless (i.e. has no atoms), $\mu(\{x\})=0$ for every $x\in S$. In fact, $\mu(X)=0$ for every finite or countably infinite set $X\subset S$. Thus every measure on a countable set has an atom (otherwise $\mu(S)$ would be $0$, contradicting the nontriviality of $\mu$). <br />
<br />
* If $\mu$ is atomless, then every set $X\subset S$ of positive measure is the disjoint union of two sets of positive measure, also, $\mu$ has a continuum of different values and if $A$ is a set of positive measure, then for every $b\in [0;\mu(A)]$, there exists $B\subset A$ such that $\mu(B)=b$.<br />
<br />
== Normal fine measures and large cardinals ==<br />
<br />
Let $\kappa$ be an [[uncountable]] [[cardinal]]. Let $\mathcal{P}_\kappa(A)$ for $|A|\geq\kappa$ be the set of all subsets of $A$ of cardinality at most $\kappa$. <br />
<br />
A filter $F$ on $\mathcal{P}_\kappa(A)$ is a ''fine filter'' if for every $a\in A$, the set $\{x\in \mathcal{P}_\kappa(A) : a\in x\}\in F$. If $F$ is also $\sigma$-complete and an ultrafilter, it is called a ''fine measure'' because it can be identified with its dual measure $\mu$ defined by $\mu(X)=1$ iff $X\in F$, $0$ otherwise.<br />
<br />
A fine measure $F$ on $\mathcal{P}_\kappa(A)$ is ''normal'' if for every function $f:\mathcal{P}_\kappa(A)\to A$, if the set $\{x\in \mathcal{P}_\kappa(A) : f(x)\in x\}\in F$ then $f$ is constant on a set in $F$, i.e. there is $k\in A$ such that $F$ also contains the set $\{x\in \mathcal{P}_\kappa(A) : f(x)=k\}$. Note that normal fine measures are also normal in the sense that they are closed under diagonal intersections, i.e. for every family $\{X_\alpha : \alpha<\kappa\}$ and $X_\alpha\in F$ for all $\alpha<\kappa$, one has $\Delta_{\alpha<\kappa}X_\alpha\in F$.<br />
<br />
If there exists a 2-valued $\kappa$-additive measure on $\kappa$, then $\kappa$ is a [[measurable]] cardinal. This equivalent to saying that there is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$. If $j:V\to\mathcal{M}$ is a nontrivial elementary embedding with critical point $\kappa$, then $\mathcal{U}_j=\{x\subset\kappa : \kappa\in j(x)\}$ is a $\kappa$-complete nonprincipal ultrafilter on $\mathcal{P}(\kappa)$ and $\kappa$ is measurable. In fact, $\mathcal{U}_j$ is a normal fine measure on $\kappa$, which we can call the "canonical" normal fine measure generated by $j$.<br />
<br />
If, for every set $S$, every $\kappa$-complete filter on $S$ can be extended to a $\kappa$-complete ultrafilter on $S$, then $\kappa$ is '''strongly compact'''. The converse is also true, every strongly compact cardinal has this property. Not that nonprincipality is not required here. Every strongly compact cardinal is measurable, and it is consistent that the first measurable and the first strongly compact cardinals are equal. Strong compactness is furthermore equivalent to the assertion that for every set $A$ such that $|A|\geq\kappa$ there exists a fine measure on $\mathcal{P}_\kappa(A)$. Those measures don't have to be normal.<br />
<br />
If there is a set $\lambda$ with $\lambda\geq\kappa$ such that there is a normal fine measure on $\mathcal{P}_\kappa(\lambda)$, then $\kappa$ is $\lambda$-[[supercompact]]; if it is $\lambda$-supercompact for every $\lambda\geq\kappa$, then it is '''supercompact'''. This is equivalent to saying that for every set $A$ with $|A|\geq\kappa$, there is a normal fine measure on $\mathcal{P}_\kappa(A)$. Clearly, every supercompact is strongly compact by the last characterization of strong compactness. It is open whether supercompactness is stronger than strong compactness consistency-wise.<br />
<br />
Every set in a normal measure is [[stationary]], also every [[measurable]] cardinal carries a normal measure containing the set of all [[inaccessible]], [[Mahlo]], and even [[Ramsey]] cardinals below it. Every [[supercompact]] cardinal $\kappa$ carries $2^{2^\kappa}$ normal measures.<br />
<br />
== See Also ==<br />
<br />
* [[Filters on N|Filters on $\mathbb{N}$]]</div>Wabb2thttp://cantorsattic.info/index.php?title=Ideal&diff=2085Ideal2017-11-11T21:42:10Z<p>Wabb2t: Redirected page to Filter</p>
<hr />
<div>#REDIRECT [[Filter]]</div>Wabb2thttp://cantorsattic.info/index.php?title=Huge&diff=2084Huge2017-11-11T21:36:51Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: Huge cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Critical points]]<br />
Huge cardinals (and their variants) were introduced by Kenneth Kunen in 1972 as a very large cardinal axiom. Kenneth Kunen first used them to prove that the consistency of the existence of a huge cardinal implies the consistency of $\text{ZFC}$+"there is a $\aleph_2$-[[filter|saturated ideal]] over $\omega_1$". <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
== Definitions ==<br />
<br />
Their formulation is similar to that of the formulation of [[superstrong]] cardinals. A huge cardinal is to a [[supercompact]] cardinal as a superstrong cardinal is to a [[strong]] cardinal, more precisely. The definition is part of a generalized phenomenon known as the "double helix", in which for some large cardinal properties $n$-$P_0$ and $n$-$P_1$, $n$-$P_0$ has less consistency strength than $n$-$P_1$, which has less consistency strength than $(n+1)$-$P_0$, and so on. This phenomenon is seen only around the [[n-fold variants|$n$-fold variants]] as of modern set theoretic concerns. <cite>Kentaro2007:DoubleHelix</cite><br />
<br />
Although they are very large, there is a first-order definition which is equivalent to $n$-hugeness, so the $\theta$-th $n$-huge cardinal is first-order definable whenever $\theta$ is first-order definable. This definition can be seen as a (very strong) strengthening of the first-order definition of [[measurable|measurability]].<br />
<br />
=== Elementary embedding definitions ===<br />
<br />
The elementary embedding definitions are somewhat standard. Let $j:V\rightarrow M$ be an [[elementary embedding]] with critical point $\kappa$ such that $M$ is a standard inner model of [[ZFC|$\text{ZFC}$]]. Then:<br />
<br />
*$\kappa$ is '''almost $n$-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length less than $\lambda$ (that is, $M^{<\lambda}\subset M$).<br />
*$\kappa$ is '''$n$-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length $\lambda$ ($M^\lambda\subset M$).<br />
*$\kappa$ is '''almost $n$-huge''' iff it is almost $n$-huge with target $\lambda$ for some $\lambda$.<br />
*$\kappa$ is '''$n$-huge''' iff it is $n$-huge with target $\lambda$ for some $\lambda$.<br />
*$\kappa$ is '''super almost $n$-huge''' iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is almost $n$-huge with target $\lambda$ (that is, the target can be made arbitrarily large).<br />
*$\kappa$ is '''super $n$-huge''' iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is $n$-huge with target $\lambda$.<br />
*$\kappa$ is '''almost huge''', '''huge''', '''super almost huge''', and '''superhuge''' iff it is '''almost $1$-huge''', '''$1$-huge''', etc. respectively.<br />
<br />
=== Ultrafilter definition ===<br />
<br />
The first-order definition of $n$-huge is somewhat similar to [[measurable|measurability]]. Specifically, $\kappa$ is measurable iff there is a nonprincipal $\kappa$-complete [[filter|ultrafilter]], $U$, over $\kappa$. A cardinal $\kappa$ is $n$-huge iff there is some cardinal $\lambda$, a nonprincipal $\kappa$-complete ultrafilter, $U$, over $\mathcal{P}(\lambda)$, and cardinals $\kappa=\lambda_0<\lambda_1<\lambda_2...<\lambda_{n-1}<\lambda_n=\lambda$ such that:<br />
<br />
$$\forall i<n\forall x\subseteq\lambda(ot(x\cap\lambda_{i+1})=\lambda_i\rightarrow x\in U)$$<br />
<br />
Where $ot(X)$ is the [[Order-isomorphism|order-type]] of the poset $(X,\in)$. <cite>Kanamori2009:HigherInfinite</cite> This definition is, more intuitively, making $U$ very large, like most ultrafilter characterizations of large cardinals ([[supercompact]], [[strongly compact]], etc.).<br />
<br />
== Consistency strength and size ==<br />
<br />
Hugeness exhibits a phenomenon associated with similarly defined large cardinals (the [[n-fold variants|$n$-fold variants]]) known as the ''double helix''. This phenomenon is when for one $n$-fold variant, letting a cardinal be called $n$-$P_0$ iff it has the property, and another variant, $n$-$P_1$, $n$-$P_0$ is weaker than $n$-$P_1$, which is weaker than $(n+1)$-$P_0$. <cite>Kentaro2007:DoubleHelix</cite> In the consistency strength hierarchy, here is where these lay (top being weakest):<br />
<br />
* [[measurable]] = $0$-[[superstrong]] = almost $0$-huge = super almost $0$-huge = $0$-huge = super $0$-huge <br />
* $n$-superstrong<br />
* $n$-fold supercompact<br />
* $(n+1)$-fold strong, $n$-fold extendible<br />
* $(n+1)$-fold Woodin, $n$-fold Vopěnka<br />
* $(n+1)$-fold Shelah<br />
* almost $n$-huge<br />
* super almost $n$-huge<br />
* $n$-huge<br />
* super $n$-huge<br />
* $(n+1)$-superstrong<br />
<br />
All huge variants lay at the top of the double helix restricted to some [[Omega|natural number]] $n$, although each are bested by [[rank-into-rank|I3]] cardinals (the [[elementary embedding|critical points]] of the I3 elementary embeddings). In fact, every I3 is preceeded by a stationary set of $n$-huge cardinals, for all $n$. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
Similarly, every huge cardinal $\kappa$ is almost huge, and there is a normal measure over $\kappa$ which contains every almost huge cardinal $\lambda<\kappa$. Every superhuge cardinal $\kappa$ is [[extendible]] and there is a normal measure over $\kappa$ which contains every extendible cardinal $\lambda<\kappa$. Every $(n+1)$-huge cardinal $\kappa$ has a normal measure which contains every cardinal $\lambda$ such that $V_\kappa\models$"$\lambda$ is super $n$-huge". <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
In terms of size, however, the least $n$-huge cardinal is smaller than the least [[supercompact]] cardinal. Assuming both exist, for any $\kappa$ which is supercompact and has an $n$-huge cardinal above it, there are $\kappa$ many $n$-huge cardinals less than $\kappa$. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
Every $n$-huge cardinal is $m$-huge for every $m\leq n$. Similarly with almost $n$-hugeness, super $n$-hugeness, and super almost $n$-hugeness. Every almost huge cardinal is [[Vopenka|Vopěnka]] (therefore the consistency of the existence of an almost-huge cardinal implies the consistency of Vopěnka's principle). <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Ultrapower&diff=2083Ultrapower2017-11-11T21:35:11Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: Ultrapower}}<br />
The intuitive idea behind ultrapower constructions (and ultraproduct constructions in general) is to take a sequence of already existing models and construct new ones from some combination of the already existing models. Ultrapower constructions are used in many major results involving elementary embeddings. A famous example is Scott's proof that the existence of a measurable cardinal implies $V\neq L$. Ultrapower embeddings are also used to characterize various large cardinal notions such as [[measurable]], [[supercompact]] and certain formulations of [[rank into rank]] embeddings. Ultrapowers have a more concrete structure than general embeddings and are often easier to work with in proofs. Most of the results in this article can be found in <cite>Jech2003:SetTheory</cite>.<br />
<br />
== General construction ==<br />
<br />
The general construction of an ultrapower supposes given an index set $X$ set for a collection of (non-empty) models $M_i$ with $i\in X$ and an [[ultrafilter]] $U$ over $X$. The ultrafilter $U$ is used to define equivalence classes over the structure $\prod_{i\in X} M_i$, the collection of all functions $f$ with domain $X$ such that $f(i)\in M_i$ for each $i\in X$. When the $M_i$ are identical to one another, we form an ''ultrapower'' by "modding out" over the equivalence classes defined by $U$. In the general case where $M_i$ differs from $M_j$, we form a structure called the ''ultraproduct'' of $\langle M_i : i\in X\rangle$.<br />
<br />
Two functions $f$ and $g$ are $U$-equivalent, denoted $f=_U g$, when the set of indices in $X$ where $f$ and $g$ agree is an element of the ultrafilter $U$ (intuitively, we think of $f$ and $g$ as disagreeing on a "small" subset of $X$). The $U$-equivalence class of $f$ is usually denoted $[f]$ and is the class of all functions $g\in \prod_{i\in X} M_i$ which are $U$-equivalent to $f$. When each $M_i$ happens to be the entire universe $V$, each $[f]$ is a proper class. To remedy this, we can use Scott's trick and only consider the members of $[f_U]$ of minimal rank to insure that $[f]$ is a set. The ultrapower (ultraproduct) is then denoted by $\text{Ult}_U(M) = M/U =\{[f]: f\in \prod_{i\in X} M_i\}$ with the membership relation defined by setting $[f]\in_U [g]$ when the set of all $i\in X$ such that $f(i)\in g(i)$ is in $U$. <br />
<br />
Note that $U$ could be a principal ultrafilter over $X$ and in this case the ultraproduct is isomorphic to almost every $M_i$, so in this case nothing new or interesting is gained by considering the ultraproduct. However, even in the case where each $M_i$ is identical and $U$ is non-principal, we have the ultrapower isomorphic to each $M_i$ but the construction technique nevertheless yields interesting results. Typical ultrapower constructions concern the case $M_i=V$. <br />
<br />
== Formal definition ==<br />
<br />
Given a collection of nonempty models $\langle M_i : i\in X \rangle$, we define the product of the collection $\langle M_i : i\in X \rangle$ as $$\prod_{i\in X}M_i = \{f:\text{dom}(f)=X \land (\forall i\in X)(f(i)\in M_i)\}$$<br />
<br />
Given an [[ultrafilter]] $U$ on $X$, we then define the following relations on $\prod_{i\in X} M_i$: Let $f,g\in\prod_{i\in X} M_i$, then $$f =_U g \iff \{i\in X : f(i)=g(i)\}\in U$$ $$f \in_U g \iff \{i\in X : f(i)\in g(i)\}\in U$$<br />
<br />
For each $f\in\prod_{i\in X} M_i$, we then define the ''equivalence class'' of $f$ in $=_U$ as follows: $$[f]=\{g: f=_U g \land \forall h(h=_U f \Rightarrow \text{rank}(g)\leq \text{rank}(h) \}$$<br />
<br />
Every member of the equivalence class of $f$ has the same rank, therefore the equivalence class is always a set, even if $M_i = V$. We now define the '''ultraproduct of $\langle M_i : i\in X \rangle$''' to be the model $\text{Ult}=(\text{Ult}_U\langle M_i : i\in X \rangle, \in_U)$ where: $$\text{Ult}_U\langle M_i : i\in X \rangle = \prod_{i\in X}M_i / U = \{[f]:f\in\prod_{i\in X}M_i\}$$<br />
<br />
If there exists a model $M$ such that $M_i=M$ for all $i\in X$, then the ultraproduct is called the '''ultrapower of $M$''', and is denoted $\text{Ult}_U(M)$.<br />
<br />
== Los' theorem ==<br />
<br />
''Los' theorem'' is the following statement: let $U$ be an ultrafilter on $X$ and $Ult$ be the ultraproduct model of some family of nonempty models $\langle M_i : i\in X \rangle$. Then, for every formula $\varphi(x_1,...,x_n)$ of set theory and $f_1,...,f_n \in \prod_{i\in X}M_i$, $$Ult\models\varphi([f_1],...,[f_n]) \iff \{i\in X : \varphi(f_1(i),...,f_n(i))\}\in U$$<br />
<br />
In particular, an ultrapower $\text{Ult}=(\text{Ult}_U(M), \in_U)$ of a model $M$ is elementarily equivalent to $M$. This is a very important result: to see why, let $f_x(i)=x$ for all $x\in M$ and $i\in X$, and now let $j_U(x)=[f_x]$ for every $x\in M$. Then $j_U$ is an [[elementary embedding]] by Los' theorem, and is called the '''canonical ultrapower embedding''' $j_U:M\to\text{Ult}_U(M)$.<br />
<br />
== Properties of ultrapowers of the universe of sets ==<br />
Let $U$ be a nonprincipal $\kappa$-complete ultrafilter on some [[measurable]] cardinal $\kappa$ and $j_U:V\to\text{Ult}_U(V)$ be the canonical ultrapower embedding of the universe. Let $\text{Ult}=\text{Ult}_U(V)$ to simplify the notation. Then:<br />
* $U\not\in\text{Ult}$<br />
* $\text{Ult}^\kappa\subseteq\text{Ult}$<br />
* $2^\kappa\leq(2^\kappa)^{\text{Ult}}<j_U(\kappa)<(2^\kappa)^+$<br />
* If $\lambda>\kappa$ is a strong limit cardinal of cofinality $\neq\kappa$ then $j(\lambda)=\lambda$.<br />
* If $\lambda$ is a limit ordinal of cofinality $\kappa$ then $j_U(\lambda)>lim_{\alpha\to\lambda}$ $j_U(\alpha)$, but if $\lambda$ has cofinality $\neq\kappa$, then $j_U(\lambda)=lim_{\alpha\to\lambda}$ $j_U(\alpha)$.<br />
<br />
Also, the following statements are equivalent:<br />
* $U$ is a normal measure<br />
* For every $X\subseteq\kappa$, $X\in U$ if and only if $\kappa\in j_U(X)$.<br />
* In $(\text{Ult}_U(V),\in_U)$, $\kappa=[d]$ where $d:\kappa\to\kappa$ is defined by $d(\alpha)=\alpha$ for every $\alpha<\kappa$.<br />
<br />
Let $j:V\to M$ be a nontrivial elementary embedding of $V$ into some transitive model $M$ with critical point $\kappa$ (which is a measurable cardinal), also let $D=\{X\subseteq\kappa:\kappa\in j(X)\}$ be the canonical normal fine measure on $\kappa$. Then:<br />
* There exists an elementary embedding $k:\text{Ult}\to M$ such that $k(j_D(x))=j(x)$ for every $x\in V$.<br />
<br />
== Iterated ultrapowers ==<br />
<br />
Given a nonprincipal $\kappa$-complete ultrafilter $U$ on some measurable cardinal $\kappa$, we define the ''iterated ultrapowers'' the following way:<br />
$$(\text{Ult}^{(0)},E^{(0)})=(V,\in)$$<br />
$$(\text{Ult}^{(\alpha+1)},E^{(\alpha+1)})=\text{Ult}_{U^{(\alpha)}}(\text{Ult}^{(\alpha)},E^{(\alpha)})$$<br />
$$(\text{Ult}^{(\lambda)},E^{(\lambda)})=lim dir_{\alpha\to\lambda}\{(\text{Ult}^{(\alpha)},E^{(\alpha)}),i_{\alpha,\beta})$$<br />
where $\lambda$ is a limit ordinal, $limdir$ denotes direct limit, $i_{\alpha,\beta} : \text{Ult}^{(\alpha)}\to \text{Ult}^{(\beta)}$ is an elementary embedding defined as follows:<br />
$$i_{\alpha,\alpha}(x)=j^{(\alpha)}(x)$$<br />
$$i_{\alpha,\alpha+n}(x)=j^{(\alpha)}(j^{(\alpha+1)}(...(j^{(\alpha+n)}(x))...))$$<br />
$$i_{\alpha,\lambda}(x)=lim_{\beta\to\lambda}i_{\alpha,\beta}(x)$$<br />
and $j^{(\alpha)}:\text{Ult}^{(\alpha)}\to \text{Ult}^{(\alpha+1)}$ is the canonical ultrapower embedding from $\text{Ult}^{(\alpha)}$ to $\text{Ult}^{(\alpha+1)}$. Also, $U^{(\alpha)}=i_{0,\alpha}(U)$ and $\kappa^{(\alpha+1)}=i_{0,\alpha}(\kappa)$ where $\kappa=\kappa^{(0)}=crit(j^{(0)})$.<br />
<br />
If $M$ is a transitive model of set theory and $U$ is (in $M$) a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, we can construct, within $M$, the iterated ultrapowers. Let us denote by $\text{Ult}^{(\alpha)}_U(M)$ the $\alpha$th iterated ultrapower, constructed in $M$.<br />
<br />
=== Properties ===<br />
<br />
* For every $\alpha$ the $\alpha$th iterated ultrapower $(\text{Ult}^{(\alpha)},E^{(\alpha)})$ is well-founded. This is due to $U$ being nonprincipal and $\kappa$-complete.<br />
<br />
* ''The Factor Lemma'': for every $\beta$, the iterated ultrapower $\text{Ult}^{(\beta)}_{U^{(\alpha)}}(\text{Ult}^{(\alpha)})$ is isomorphic to the iterated ultrapower $\text{Ult}^{(\alpha+\beta)}$.<br />
<br />
* For every limit ordinal $\lambda$, $\text{Ult}^{(\lambda)}\subseteq \text{Ult}^{(\alpha)}$ for every $\alpha<\lambda$. Also, $\kappa^{(\lambda)}=lim_{\alpha\to\lambda}$ $\kappa^{(\alpha)}$.<br />
<br />
* For every $\alpha$, $\beta$ such that $\alpha>\beta$, one has $\kappa^{(\alpha)}>\kappa^{(\beta)}$.<br />
<br />
* If $\gamma<\kappa^{(\alpha)}$ then $i_{\alpha,\beta}(\gamma)=\gamma$ for all $\beta\geq\alpha$.<br />
<br />
* If $X\subseteq\kappa^{(\alpha)}$ and $X\in \text{Ult}^{(\alpha)}$ then for all $\beta\geq\alpha$, one has $X=\kappa^{(\alpha)}\cap i_{\alpha,\beta}(X)$.<br />
<br />
=== The representation lemma ===<br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Measurable&diff=2081Measurable2017-11-11T21:31:16Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: Measurable cardinal}}<br />
A '''measurable cardinal''' $\kappa$ is an [[uncountable]] [[cardinal]] such that it is possible to "measure" the subsets of $\kappa$ using a 2-valued [[measure]] on the powerset of $\kappa$, $\mathcal{P}(\kappa)$. There exists several other equivalent definitions: For example, $\kappa$ can also be the critical point of a nontrivial [[elementary embedding]] $j:V\to M$.<br />
<br />
Every measurable is a large cardinal, i.e. $V_\kappa$ satisfies $\text{ZFC}$, therefore $\text{ZFC}$ cannot prove the existence of a measurable cardinal. In fact $\kappa$ is [[inaccessible]], the $\kappa$th inacessible, the $\kappa$th [[weakly compact]] cardinal, the $\kappa$th [[Ramsey]], and similarly bears most of the large cardinal properties under Ramsey-ness. It is notable that every measurable has the mentioned properties in $\text{ZFC}$, but in $\text{ZF}$ they may not (but their existence remains consistency-wise ''much'' stronger than existence of cardinals with those properties), in fact under the [[axiom of determinacy]], the first two uncountable cardinals, $\aleph_1$ and $\aleph_2$, are both measurable.<br />
<br />
Measurable cardinals were introduced by Stanislaw Ulam in 1930.<br />
<br />
== Definitions ==<br />
<br />
The following definitions are equivalent for every uncountable cardinal $\kappa$:<br />
# There exists a 2-valued measure on $\kappa$.<br />
# There exists a $\kappa$-complete (or even just $\sigma$-complete) nonprincipal ultrafilter on $\kappa$.<br />
# There exists a nontrivial elementary embedding $j:V\to M$ with $M$ a transitive class and such that $\kappa$ is the least ordinal moved (the ''critical point'').<br />
# There exists an ultrafilter $U$ on $\kappa$ such that the [[ultrapower]] $(\text{Ult}_U(V),\in_U)$ of the universe is well-founded and isn't isomorphic to $V$.<br />
<br />
The equivalence between the first two definition is due to the fact that if $\mu$ is a 2-valued measure on $\kappa$, then $U=\{X\subset\kappa|\mu(X)=1\}$ is a nonprincipal ultrafilter (since $\mu$ is 2-valued) and is also $\sigma$-complete because of $\mu$'s $\sigma$-additivity. Similarly, if $U$ is a $\sigma$-complete nonprincipal ultrafilter on $\kappa$, then $\mu:\mathcal{P}(\kappa)\to[0,1]$ defined by $\mu(X)=1$ whenever $X\in U$, $\mu(X)=0$ otherwise is a 2-valued measure on $\kappa$.<br />
<br />
To see that the third definition implies the first two, one can show that if $j:V\to M$ is a nontrivial elementary embedding, then the set $\mathcal{U}=\{x\subset\kappa|\kappa\in j(x)\})$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, and in fact a normal fine measure. To show the converse, one needs to use [[ultrapower|ultrapower embeddings]]: if $U$ is a nonprincipal $\kappa$-complete ultrafilter on $\kappa$, then the canonical ultrapower embedding $j:V\to\text{Ult}_U(V)$ is a nontrivial elementary embedding of the universe.<br />
<br />
The equivalence of the last definition with the other ones is simply due to the fact that the ultrapower $(\text{Ult}_U(V),\in_U)$ of the universe is well-founded if and only if $U$ is $\sigma$-complete, and is isomorphic to $V$ if and only if $U$ is principal.<br />
<br />
== Properties ==<br />
<br />
''See also: [[Ultrapower]]''<br />
<br />
If $\kappa$ is measurable, then it has a measure that take every value in $[0,1]$. Also there must be a normal fine measure on $\mathcal{P}_\kappa(\kappa)$.<br />
<br />
Every measurable cardinal is [[regular]], and (under AC) bears most large cardinal properties weaker than it. It is in particular $\Pi^2_1$-[[indescribable]]. However the least measurable cardinal is not $\Sigma^2_1$-indescribable. Independently of the truth of AC, the existence of a measurable cardinal implies the consistency of the existence of large cardinals with the said properties, even if that measurable is merely $\omega_1$.<br />
<br />
If $\kappa$ is measurable and $\lambda<\kappa$ then it cannot be true that $\kappa<2^\lambda$. Under AC this means that $\kappa$ is a strong limit (and since it is regular, it must be strongly inaccessible, hence it cannot be $\omega_1$).<br />
<br />
If there exists a measurable cardinal then [[zero sharp|$0^\#$]] exists, and therefore $V\neq L$. In fact, the [[zero sharp|sharp]] of every real number exists, and therefore $\mathbf{\Pi}^1_1$-[[axiom of determinacy|determinacy]] holds. Furthermore, assuming the axiom of determinacy, the cardinals $\omega_1$, $\omega_2$, $\omega_{\omega+1}$ and $\omega_{\omega+2}$ are measurable, also in $L(\mathbb{R})$ every regular cardinal smaller than [[theta|$\Theta$]] is measurable.<br />
<br />
Every measurable has the following reflection property: let $j:V\to M$ be a nontrivial elementary embedding with critical point $\kappa$. If $x\in V_\kappa$ and $M\models\varphi(\kappa,x)$ for some first-order formula $\varphi$, then the set of all ordinals $\alpha<\kappa$ such that $V\models\varphi(\alpha,x)$ is [[stationary]] in $\kappa$ and has the same measure as $\kappa$ itself by any 2-valued measure on $\kappa$.<br />
<br />
Measurability of $\kappa$ is equivalent with $\kappa$-strong compactness of $\kappa$, and also with $\kappa$-supercompactness of $\kappa$ (fragments of [[strongly compact | strong compactness]] and [[supercompact | supercompactness]] respectively.) It is also consistent with $\text{ZFC}$ that the first measurable cardinal and the first [[strongly compact]] cardinal are equal.<br />
<br />
If a measurable $\kappa$ is such that there is $\kappa$ [[strongly compact]] cardinals below it, then it is strongly compact. If it is a limit of strongly compact cardinals, then it is strongly compact yet not [[supercompact]]. If a measurable $\kappa$ has infinitely many [[Woodin]] cardinals below it, then the axiom of determinacy holds in $L(\mathbb{R})$, also the [[axiom of projective determinacy]] holds.<br />
<br />
If $\kappa$ is measurable in a ground model, then it is measurable in any forcing extension of that ground model whose notion of forcing has cardinality strictly smaller than $\kappa$. Prikry showed however that every measurable can be collapsed to a cardinal of cofinality $\omega$ and no other cardinal is collapsed.<br />
<br />
=== Failure of $\text{GCH}$ at a measurable ===<br />
<br />
Gitik proved that the following statements are equiconsistent:<br />
* The generalized continuum hypothesis fails at a measurable cardinal $\kappa$, i.e. $2^\kappa > \kappa^+$<br />
* The singular cardinal hypothesis fails, i.e. there is a strong limit singular $\kappa$ such that $2^\kappa > \kappa^+$<br />
* There is a measurable cardinal of [[Mitchell rank | Mitchell order]] $\kappa^{++}$, i.e. $o(\kappa)=\kappa^{++}$<br />
<br />
Thus violating $\text{GCH}$ at a measurable (or violating the SCH at any strong limit cardinal) is strictly stronger consistency-wise than the existence of a measurable cardinal.<br />
<br />
However, if the generalized continuum hypothesis fails at a measurable, then it fails at $\kappa$ many cardinals below it.<br />
<br />
== Real-valued measurable cardinal ==<br />
<br />
A cardinal $\kappa$ is '''real-valued''' measurable if there exists a $\kappa$-additive measure on $\kappa$. The smallest cardinal $\kappa$ carrying a $\sigma$-additive 2-valued measure must also carry a $\kappa$-additive measure, and is therefore real-valued measurable, also it is strongly inaccessible under AC.<br />
<br />
If a real-valued measurable cardinal is not measurable, then it must be smaller than (or equal to) $2^{\aleph_0}$. [[Martin's axiom]] implies that the continuum is not real-valued measurable.<br />
<br />
Solovay showed that the existence of a measurable cardinal is equiconsistent with the existence of a real-valued measurable cardinal. More precisely, he showed that if there is a measurable then there is generic extension in which $\kappa=2^{\aleph_0}$ and $\kappa$ is real-valued measurable, and conversely if there exists a real-valued measurable then it is measurable in some model of $\text{ZFC}$.<br />
<br />
== See also ==<br />
* [[Ultrapower]]<br />
* [[Mitchell order]]<br />
* [[Axiom of determinacy]]<br />
* [[Strongly compact]] cardinal<br />
<br />
== Read more ==<br />
* Jech, Thomas - ''Set theory''<br />
<br />
* Bering A., Edgar - ''A brief introduction to measurable cardinals''</div>Wabb2thttp://cantorsattic.info/index.php?title=ORD_is_Mahlo&diff=2080ORD is Mahlo2017-11-11T21:28:44Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: $\text{Ord}$ is Mahlo}}<br />
The assertion ''$\text{Ord}$ is Mahlo'' is the scheme expressing that the proper class [[REG]] consisting of all regular cardinals is a [[stationary]] proper class, meaning that it has elements from every definable (with parameters) [[closed unbounded]] proper class of ordinals. In other words, the scheme asserts for every formula $\varphi$, that if for some parameter $z$ the class $\{\alpha\mid \varphi(\alpha,z)\}$ is a closed unbounded class of ordinals, then it contains a regular cardinal.<br />
<br />
* If $\kappa$ is [[Mahlo]], then $V_\kappa\models\text{Ord is Mahlo}$. <br />
* Consequently, the existence of a Mahlo cardinal implies the consistency of $\text{Ord}$ is Mahlo, and the two notions are not equivalent.<br />
* Moreoever, since the ORD is Mahlo scheme is expressible as a first-order theory, it follows that whenever $V_\gamma\prec V_\kappa$, then also $V_\gamma$ satisfies the Levy scheme. <br />
* Consequently, if there is a Mahlo cardinal, then there is a club of cardinals $\gamma\lt\kappa$ for which $V_\gamma\models\text{Ord is Mahlo}$. <br />
<br />
A simple compactness argument establishes that $\text{Ord}$ is Mahlo is equiconsistent over $\text{ZFC}$ with the existence of an [[inaccessible reflecting cardinal]]. On the one hand, if $\kappa$ is an inaccessible reflecting cardinal, then since $V_\kappa\prec V$ it follows that any class club definable in $V$ with parameters below $\kappa$ will be unbounded in $\kappa$ and hence contain $\kappa$ as an element and consequently contain an inaccessible cardinal. On the other hand, if $\text{Ord}$ is Mahlo is consistent, then every finite fragment of the theory asserting that $\kappa$ is an inaccessible reflecting cardinal (which is after all asserted as a scheme) is consistent, and hence by compactness the whole theory is consistent.</div>Wabb2thttp://cantorsattic.info/index.php?title=Second-order&diff=2079Second-order2017-11-11T21:27:19Z<p>Wabb2t: </p>
<hr />
<div>'''Morse Kelley''' (commonly abbreviated as ''$\text{MK}$'' or ''$\text{KM}$'') is a first-order countably infinite axiomatic theory which is stronger than [[ZFC|$\text{ZFC}$]] in consistency strength. It was named after John L. Kelley and Anthony Morse. It is very similar to the second-order form of [[ZFC]|$\text{ZFC}$] (known as [[$\text{NBG}$]]), although slightly strengthened with a modification of class comprehension.<br />
<br />
In $\text{MK}$, the sets are precisely the classes which are in another class. $X$ is a set iff $\exists W(X\in W)$. This is often abbreviated $MX$.<br />
<br />
== Axioms ==<br />
<br />
The axioms of $\text{MK}$ are exactly those of [[NBG|$\text{NBG}$]] except with a modified '''Axiom Schema of Class Comprehension'''. This modified version allows classes to be defined using other classes, as long as none of them are that class. Specifically, for any $\phi$ and any $n$:<br />
<br />
$$\forall W_1\forall W_2...\forall W_n(\neg\forall x(\phi(x,W_1...W_n)\Leftrightarrow x\in W_1)\land \neg\forall x(\phi(x,W_1...W_n)\Leftrightarrow x\in W_2)...\neg\forall x(\phi(x,W_1...W_n)\Leftrightarrow x\in W_n)$$<br />
$$\rightarrow\exists Y(\forall x(\phi(x,W_1...W_n)\Leftrightarrow x\in Y))$$<br />
<br />
In other words, for any formula $\phi$ with variables $x$ and $W_1,...,W_n$, there is a class $Y=\{x:\phi(x,W_1...W_n)\}$ as long as no $W_m=Y$ and $Y$ does not occur in $\phi$. This is because, if we allowed $Y$ to occur in $\phi$ or if we allowed $Y$ to be passed as a parameter for $\phi$, then we could create a sentence $\phi$ such that $\phi(x,W_1,...,W_n)\iff x\not\in Y$, and then we would have $\forall x(x\in Y\iff x\not\in Y)$, a contradiction. This is generally considered to be a useful definition for a class, allowing almost all known "useful" classes to be created.<br />
<br />
The other most important axiom (which is also in $\text{NBG}$) is the '''Axiom of Limitation of Size''', which asserts that a class is not a set iff it has a bijection onto $V$. This is a particularly strong axiom, implying the [[Axiom of Choice]]. Furthermore, it implies that every class can be well-ordered (known as the axiom of global choice). <br />
<br />
== Models of $\text{MK}$ ==<br />
<br />
In consistency strength, $\text{MK}$ is stronger than [[ZFC|$\text{ZFC}$]] and weaker than the [[positive set theory]] $\text{GPK}^{+}_\infty$. It directly implies the consistency of $\text{ZFC}$. However, if a cardinal $\kappa$ is worldly then $V_{\kappa+1}\models\text{MK}$. <br />
<br />
Because it uses classes, set models of $\text{MK}$ are generally taken to be the powerset of some model of $\text{ZFC}$. For this reason, a large cardinal axiom with $V_\kappa$ having some elementary property can be strengthened by instead using $V_{\kappa+1}$. When doing this with [[indescribable|$\Pi_m^0$-indescribability]], one achieves [[indescribable|$\Pi_m^1$-indescribability]] (which is considerably stronger). When doing this with [[extendible|$0$-extendibility]] (which is equiconsistent with $\text{ZFC}$), one achieves [[extendible|$1$-extendibility]] (which is so much stronger that it actually implies the consistency of a [[supercompact]] cardinal).</div>Wabb2thttp://cantorsattic.info/index.php?title=Positive_set_theory&diff=2078Positive set theory2017-11-11T21:24:20Z<p>Wabb2t: </p>
<hr />
<div>''Positive set theory'' is the name of a certain group of axiomatic set theories originally created as an example of a (nonstandard) set theories in which the axiom of foundation fails (e.g. there exists $x$ such that $x\in x$). <cite>FortiHinnion89:ConsitencyProblemPositiveComp</cite> Those theories are based on a weakening of the (inconsistent) ''comprehension axiom'' of [[naive set theory]] (which asserts that every formula $\phi(x)$ defines a set that contains all $x$ such that $\phi(x)$) by restraining the formulas used to a smaller class of formulas called ''positive'' formulas.<br />
<br />
While most positive set theories are weaker than [[ZFC|$\text{ZFC}$]] (and usually mutually interpretable with [[:wikipedia:second-order arithmetic|second-order arithmetic]]), one of them, $\text{GPK}^+_\infty$ turns out to be very powerful, being mutually interpretable with [[Morse-Kelley set theory]] plus an axiom asserting that the class of all [[ordinal|ordinals]] is [[weakly compact]]. <cite>Esser96:InterpretationZFCandMKinPositiveTheory</cite><br />
<br />
== Positive formulas ==<br />
<br />
In the first-order language $\{=,\in\}$, we define a ''BPF formula'' (bounded positive formula) the following way <cite>Esser96:InterpretationZFCandMKinPositiveTheory</cite>:<br />
For every variable $x$, $y$ and BPF formulas $\varphi$, $\psi$,<br />
* $x=y$ and $x\in y$ are BPF.<br />
* $\varphi\land\psi$, $\varphi\lor\psi$, $\exists x\varphi$ and $(\forall x\in y)\varphi$ are BPF.<br />
<br />
A formula is then a ''GPF formula'' (generalized positive formula) if it is a BPF formula or if it is of the form $\forall x(\theta(x)\Rightarrow\varphi)$ with $\theta(x)$ a GPF formula with exactly one free variable $x$ and no parameter and $\varphi$ is a GPF formula (possibly with parameters). <cite>Esser96:GPKAFA</cite><br />
<br />
== $\text{GPK}$ positive set theories ==<br />
<br />
The positive set theory $\text{GPK}$ consists of the following axioms:<br />
* '''Empty set''': $\exists x\forall y(y\not\in x)$.<br />
* '''Extensionality''': $\forall x\forall y(x=y\Leftrightarrow\forall z(z\in x\Leftrightarrow z\in y))$.<br />
* '''GPF comprehension''': the universal closure of $\exists x\forall y(y\in x\Leftrightarrow\varphi)$ for every GPF formula $\varphi$ (with parameters) in which $x$ does not occur.<br />
The empty set axiom is necessary, as without it the theory would hold in the trivial model which has only one element satisfying $x=\{x\}$. Note that, while $\text{GPK}$ do proves the existence of a set such that $x\in x$, Olivier Esser proved that it refutes the [[:wikipedia:anti-foundation axiom|anti-foundation axiom]] (AFA). <cite>Esser96:GPKAFA</cite><br />
<br />
The theory $\text{GPK}^+$ is obtained by adding the following axiom:<br />
* '''Closure''': the universal closure of $\exists x(\forall z(\varphi(z)\Rightarrow z\in x)\land\forall y(\forall w(\varphi(z)\Rightarrow z\in y)\Rightarrow y\subset x))$ for every formula $\varphi(z)$ (not necessarily BPF or GPF) with a free variable $z$ (and possibly parameters) such that $x$ does not occur in $\varphi$.<br />
This axiom scheme asserts that for any (possibly proper) class $C=\{x|\varphi(x)\}$ there is a smallest set $X$ containing $C$, i.e. $C\subset X$ and for all sets $Y$ such that $C\subset Y$, one has $X\subset Y$. <cite>Esser99:ConsistencyPositiveTheory</cite><br />
<br />
Note that replacing GPF comprehension in $\text{GPK}^+$ by BPF comprehension does not make the theory any weaker: BPF comprehension plus Closure implies GPF comprehension.<br />
<br />
Both $\text{GPK}$ and $\text{GPK}^+$ are consistent relative to $\text{ZFC}$, in fact mutually interpretable with second-order arithmetic. However a much stronger theory, '''$\text{GPK}^+_\infty$''', is obtained by adding the following axiom:<br />
* '''Infinity''': the von Neumann ordinal $\omega$ is a set.<br />
By "von Neumann ordinal" we mean the usual definition of ordinals as well-ordered-by-inclusion sets containing all the smaller ordinals. Here $\omega$ is the set of all finite ordinals (the natural numbers). The point of this axiom is not implying the existence of an infinite set; the ''class'' $\omega$ exists, so it has a set closure which is certainely infinite. This set closure happens to satisfy the usual axiom of infinity of $\text{ZFC}$ (i.e. it contains 0 and the successor of all its members) but in $\text{GPK}^+$ this is not enough to deduce that $\omega$ itself is a set (an improper class).<br />
<br />
Olivier Esser showed that $\text{GPK}^+_\infty$ can not only interpret $\text{ZFC}$ (and prove it consistent), but is in fact mutually interpretable with a ''much'' stronger set theory, namely, Morse-Kelley set theory with an axiom asserting that the (proper) class of all ordinals is [[weakly compact]]. This theory is powerful enough to prove, for instance, that there exists a proper class of [[Mahlo|hyper-Mahlo]] cardinals. <cite>Esser96:InterpretationZFCandMKinPositiveTheory</cite><br />
<br />
== As a topological set theory ==<br />
''To be expanded.''<br />
== The axiom of choice and $\text{GPK}$ set theories ==<br />
''To be expanded. <cite>Esser2000:InconsistencyACwithGPK</cite><cite>FortiHinnion89:ConsitencyProblemPositiveComp</cite>''<br />
== Other positive set theories and the inconsistency of the axiom of extensionality ==<br />
''To be expanded. <cite>Esser99:ExtensionalityInPositiveTheory</cite>''<br />
<br />
{{References}}<br />
<br />
{{stub}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Wholeness_axioms&diff=2077Wholeness axioms2017-11-11T21:22:06Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: The Wholeness Axioms}}<br />
The wholeness axioms, proposed by Paul Corazza <cite>Corazza2000:WholenessAxiomAndLaverSequences, Corazza2003:GapBetweenI3andWA</cite>, occupy a<br />
high place in the upper stratosphere of the large cardinal<br />
hierarchy, intended as slight weakenings of the [[Kunen inconsistency]], but similar in spirit. <br />
<br />
The ''wholeness axioms'' are formalized in the<br />
language $\{\in,j\}$, augmenting the usual language of set<br />
theory $\{\in\}$ with an additional unary function symbol $j$<br />
to represent the [[elementary embedding]]. The base theory ZFC is<br />
expressed only in the smaller language $\{\in\}$. Corazza's<br />
original proposal, which we denote by $\text{WA}_0$, asserts<br />
that $j$ is a nontrivial amenable elementary embedding<br />
from the universe to itself. Elementarity is expressed by<br />
the scheme $\varphi(x)\iff\varphi(j(x))$, where $\varphi$<br />
runs through the formulas of the usual language of set<br />
theory; nontriviality is expressed by the sentence $\exists<br />
x j(x)\not=x$; and amenability is simply the assertion<br />
that $j\upharpoonright A$ is a set for every set $A$. Amenability in this case is equivalent to<br />
the assertion that the separation axiom holds for<br />
$\Delta_0$ formulae in the language $\{\in,j\}$. <br />
The wholeness axiom $\text{WA}$, also denoted $\text{WA}_\infty$, asserts in addition that the<br />
full separation axiom holds in the language $\{\in,j\}$. <br />
<br />
Those two axioms are the endpoints of the hierarchy of axioms $\text{WA}_n$, asserting increasing amounts of the separation axiom. <br />
Specifically, the wholeness axiom $\text{WA}_n$, where $n$ is<br />
amongst $0,1,\ldots,\infty$, consists of the following:<br />
<br />
# (elementarity) All instances of $\varphi(x)\iff\varphi(j(x))$ for $\varphi$ in the language $\{\in,j\}$.<br />
# (separation) All instances of the Separation Axiom for $\Sigma_n$ formulae in the full language $\{\in,j\}$.<br />
# (nontriviality) The axiom $\exists x\,j(x)\not=x$.<br />
<br />
Clearly, this resembles the [[Kunen inconsistency]]. What is missing from the wholeness<br />
axiom schemes, and what figures prominantly in Kunen's<br />
proof, are the instances of the replacement axiom in the<br />
full language with $j$. In particular, it is the replacement axiom in the language with $j$ that allows one to define the critical sequence $\langle \kappa_n\mid n\lt\omega\rangle$, where $\kappa_{n+1}=j(\kappa_n)$, which figures in all the proofs of the Kunen inconsistency. Thus, none of the proofs of the Kunen inconsistency can be carried out with WA, and indeed, in every model of $\text{WA}$ the critical sequence is unbounded in the ordinals. <br />
<br />
The hiearchy of wholeness axioms is strictly increasing in strength, if consistent. <cite>Hamkins2001:WholenessAxiomAndVequalHOD</cite> <br />
<br />
If $j:V_\lambda\to V_\lambda$ witnesses a [[rank into rank]] cardinal, then $\langle V_\lambda,\in,j\rangle$ is a model of the wholeness axiom.<br />
<br />
If the wholeness axiom is consistent with $\text{ZFC}$, then it is consistent with $\text{ZFC+V=HOD}$.<cite>Hamkins2001:WholenessAxiomAndVequalHOD</cite><br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=L_of_V_lambda%2B1&diff=2076L of V lambda+12017-11-11T21:20:45Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: $\exists j:L(V_{\lambda+1})\to L(V_{\lambda+1})$}}<br />
The large cardinal axiom of the title asserts that some non-trivial elementary embedding $j:V_{\lambda+1}\to V_{\lambda+1}$ extends to a non-trivial elementary embedding $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$, where $L(V_{\lambda+1})$ is the transitive proper class obtained by starting with $V_{\lambda+1}$ and forming the constructible hierarchy over $V_{\lambda+1}$ in the usual fashion. An alternate, but equivalent formulation asserts the existence of some non-trivial elementary embedding $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$ with $cr(j) < \lambda$. The critical point assumption is essential for the large cardinal strength as otherwise the axiom would follow from the existence of some measurable cardinal above $\lambda$. The axiom is of [[rank into rank]] type, despite its formulation as an embedding between proper classes, and embeddings witnessing the axiom known as $\text{I0}$ embeddings.<br />
<br />
Originally formulated by Woodin in order to establish the relative consistency of a strong [[determinacy]] hypothesis, it is now known to be obsolete for this purpose (it is far stronger than necessary). Nevertheless, research on the axiom and its variants is still widely pursued and there are numerous intriguing open questions regarding the axiom and its variants, see . <br />
<br />
The axiom subsumes a hierarchy of the strongest large cardinals not known to be inconsistent with $\text{ZFC}$ and so is seen as ``straining the limits of consistency" <cite>Kanamori2009:HigherInfinite</cite>. An immediate observation due to the [[Kunen inconsistency]] is that, under the $\text{I0}$ axiom, $L(V_{\lambda+1})$ ''cannot'' satisfy the axiom of choice. <br />
<br />
==The $L(V_{\lambda+1})$ Hierarchy==<br />
<br />
==Relation to the I1 Axiom==<br />
<br />
==Ultrapower Reformulation==<br />
<br />
==Similarities with $L(\mathbb{R})$ under Determinacy==<br />
<br />
==Strengthenings of $\text{I0}$ and Woodin's $E_\alpha(V_{\lambda+1})$ Sequence==<br />
<br />
{{References}}<br />
<br />
{{stub}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Rank_into_rank&diff=2075Rank into rank2017-11-11T21:19:53Z<p>Wabb2t: Undo revision 2052 by Wabb2t (talk)</p>
<hr />
<div>{{DISPLAYTITLE: Rank into rank}}<br />
A nontrivial [[elementary embedding]] $j:V_\lambda\to V_\lambda$ for some infinite ordinal $\lambda$ is known as a ''rank into rank embedding'' and the axiom asserting that such an embedding exists is usually denoted by $\text{I3}$, $\text{I2}$, $\text{I1}$, $\mathcal{E}(V_\lambda)\neq \emptyset$ or some variant thereof. The term applies to a hierarchy of such embeddings increasing in large cardinal strength reaching toward the [[Kunen inconsistency]]. The axioms in this section are in some sense a technical restriction falling out of Kunen's proof that there can be no nontrivial elementary embedding $j:V\to V$ in $\text{ZFC}$). An analysis of the proof shows that there can be no nontrivial $j:V_{\lambda+2}\to V_{\lambda+2}$ and that if there is some ordinal $\delta$ and nontrivial rank to rank embedding $j:V_\delta\to V_\delta$ then necessarily $\delta$ must be a strong limit cardinal of cofinality $\omega$ or the successor of one. By standing convention, it is assumed that rank into rank embeddings are not the identity on their domains.<br />
<br />
There are really two cardinals relevant to such embeddings: The large cardinal is the critical point of $j$, often denoted $crit(j)$ or sometimes $\kappa_0$, and the other (not quite so large) cardinal is $\lambda$. In order to emphasize the two cardinals, the axiom is sometimes written as $E(\kappa,\lambda)$ (or $\text{I3}(\kappa,\lambda)$, etc.) as in <cite>Kanamori2009:HigherInfinite</cite>. The cardinal $\lambda$ is determined by defining the ''critical sequence of $j$''. Set $\kappa_0 = crit(j)$ and $\kappa_{n+1}=j(\kappa_n)$. Then $\lambda = \sup \langle \kappa_n : n <\omega\rangle$ and is the first fixed point of $j$ that occurs above $\kappa_0$. Note that, unlike many of the other large cardinals appearing in the literature, the ordinal $\lambda$ is ''not'' the target of the critical point; it is the $\omega^{th}$ $j$-iterate of the critical point. <br />
<br />
As a result of the strong closure properties of rank into rank embeddings, their critical points are [[huge]] and in fact $n$-huge for every $n$. This aspect of the large cardinal property is often called $\omega$-hugeness and the term ''$\omega$-huge cardinal'' is sometimes used to refer to the critical point of some rank into rank embedding. <br />
<br />
==The $\text{I3}$ Axiom and Natural Strengthenings== <br />
<br />
The $\text{I3}$ axiom asserts, generally, that there is some embedding $j:V_\lambda\to V_\lambda$. $\text{I3}$ is also denoted as $\mathcal{E}(V_\lambda)\neq\emptyset$ where $\mathcal{E}(V_\lambda)$ is the set of all elementary embeddings from $V_\lambda$ to $V_\lambda$, or sometimes even $\text{I3}(\kappa,\lambda)$ when mention of the relevant cardinals is necessary. In its general form, the axiom asserts that the embedding preserves all first-order structure but fails to specify how much second-order structure is preserved by the embedding. The case that ''no'' second-order structure is preserved is also sometimes denoted by $\text{I3}$. In this specific case $\text{I3}$ denotes the weakest kind of rank into rank embedding and so the $\text{I3}$ notation for the axiom is somewhat ambiguous. To eliminate this ambiguity we say $j$ is $E_0(\lambda)$ when $j$ preserves only first-order structure. <br />
<br />
The axiom can be strengthened and refined in a natural way by asserting that various degrees of second-order correctness are preserved by the embeddings. A rank into rank embedding $j$ is said to be $\Sigma^1_n$ or ''$\Sigma^1_n$ correct'' if, for every $\Sigma^1_n$ formula $\Phi$ and $A\subseteq V_\lambda$ the elementary schema holds for $j,\Phi$, and $A$: $$V_\lambda\models\Phi(A) \Leftrightarrow V_\lambda\models\Phi(j(A)).$$<br />
The more specific axiom $E_n(\lambda)$ asserts that some $j\in\mathcal{E}(V_\lambda)$ is $\Sigma^1_{2n}$. <br />
<br />
The ``$2n$" subscript in the axiom $E_n(\lambda)$ is incorporated so that the axioms $E_m(\lambda)$ and $E_n(\lambda)$ where $m<n$ are strictly increasing in strength. This is somewhat subtle. For $n$ odd, $j$ is $\Sigma^1_n$ if and only if $j$ is $\Sigma^1_{n+1}$. However, for $n$ even, $j$ being $\Sigma^1_{n+1}$ is ''significantly'' stronger than a $j$ being $\Sigma^1_n$<cite>Laver1997:Implications</cite>. <br />
<br />
==The $\text{I2}$ Axiom==<br />
<br />
Any $j:V_\lambda\to V_\lambda$ can be extended to a $j^+:V_{\lambda+1}\to V_{\lambda+1}$ but in only one way: Define for each $A\subseteq V_\lambda$ $$j^+(A)=\bigcup_{\alpha < \lambda}(j(V_\alpha\cap A)).$$ $j^+$ is not necessarily elementary. The $\text{I2}$ axiom asserts the existence of some elementary embedding $j:V\to M$ with $V_\lambda\subseteq M$ where $\lambda$ is defined as the $\omega^{th}$ $j$-iterate of the critical point. Although this axiom asserts the existence of a ''class'' embedding with a very strong closure property, it is in fact equivalent to an embedding $j:V_\lambda\to V_\lambda$ with $j^+$ preserving well-founded relations on $V_\lambda$. So this axioms preserves ''some'' second-order structure of $V_\lambda$ and is in fact equivalent to $E_1(\lambda)$ in the hierarchy defined above. A specific property of $\text{I2}$ embeddings is that they are ''iterable'' (i.e. the direct limit of directed system of embeddings is well-founded). In the literature, $IE(\lambda)$ asserts that $j:V_\lambda\to V_\lambda$ is iterable and $IE(\lambda)$ falls strictly between $E_0(\lambda)$ and $E_1(\lambda)$. <br />
<br />
As a result of the strong closure property of $\text{I2}$, the equivalence mentioned above cannot be through an analysis of some ultrapower embedding. Instead, the equivalence is established by constructing a directed system of embeddings of various ultrapowers and using reflection properties of the critical points of the embeddings. The direct limit is well-founded since well-founded relations are preserved by $j^+$. The use of both direct and indirect limits, in conjunction with reflection arguments, is typical for establishing the properties of rank into rank embeddings. <br />
<br />
==The $\text{I1}$ Axiom==<br />
<br />
$\text{I1}$ asserts the existence of a nontrivial elementary embedding $j:V_{\lambda+1}\to V_{\lambda+1}$. This axiom is sometimes denoted $\mathcal{E}(V_{\lambda+1})\neq\emptyset$. Any such embedding preserves all second-order properties of $V_\lambda$ and so is $\Sigma^1_n$ for all $n$. To emphasize the preservation of second-order properties, the axiom is also sometimes written as $E_\omega(\lambda)$. In this case, restricting the embedding to $V_\lambda$ and forming $j^+$ as above yields the original embedding. <br />
<br />
Strengthening this axiom in a natural way leads the $\text{I0}$ axiom, i.e. asserting that embeddings of the form [[L of V lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]] exist. There are also other natural strengthenings of $\text{I1}$, though it is not entirely clear how they relate to the $\text{I0}$ axiom. For example, one can assume the existence of elementary embeddings satisfying $\text{I1}$ which extend to embeddings $j:M\to M$ where $M$ is a transitive class inner model and add various requirements to $M$. These requirements must not entail that $M$ satisfies the axiom of choice by the Kunen inconsistency. Requirements that have been considered include assuming $M$ contains $V_{\lambda+1}$, $M$ satisfies $DC_\lambda$, $M$ satisfies replacement for formulas containing $j$ as a parameter, $j(crit(j))$ is arbitrarily large in $M$, etc. <br />
<br />
==Large Cardinal Properties of Critical Points==<br />
<br />
The critical points of rank into rank embeddings have many strong reflection properties. They are measurable, $n$-huge for all $n$ (hence the terminology $\omega$-huge mentioned in the introduction) and partially supercompact. <br />
<br />
Using $\kappa_0$ as a seed, one can form the ultrafilter $$U=\{X\subseteq\mathcal{P}(\kappa_0): j"\kappa_0\in j(X)\}.$$ Thus, $\kappa_0$ is a measurable cardinal.<br />
<br />
In fact, for any $n$, $\kappa_0$ is also $n$-huge as witnessed by the ultrafilter <br />
$$U=\{X\subseteq\mathcal{P}(\kappa_n): j"\kappa_n\in j(X)\}.$$ This motivates the term $\omega$-huge cardinal mentioned in the introduction. <br />
<br />
Letting $j^n$ denote the $n^{th}$ iteration of $j$, then <br />
$$V_\lambda\models ``\lambda\text{ is supercompact"}.$$ To see this, suppose $\kappa_0\leq \theta <\kappa_n$, then $$U=\{X\subseteq\mathcal{P}_{\kappa_0}(\theta): j^n"\theta\in j^n(X)\}$$ winesses the $\theta$-compactness of $\kappa_0$ (in $V_\lambda$). For this last claim, it is enough that $\kappa_0(j)$ is $<\lambda$-supercompact, i.e. not *fully* supercompact in $V$. In this case, however, $\kappa_0$ *could* be fully supercompact. <br />
<br />
Critical points of rank-into-rank embeddings also exhibit some *upward* reflection properties. For example, if $\kappa$ is a critical point of some embedding witnessing $\text{I3}(\kappa,\lambda)$, then there must exist another embedding witnessing $\text{I3}(\kappa',\lambda)$ with critical point ''above'' $\kappa$! This upward type of reflection is not exhibited by large cardinals below [[extendible]] cardinals in the large cardinal hierarchy.<br />
<br />
==Algebras of elementary embeddings==<br />
<br />
If $j,k\in\mathcal{E}_{\lambda}$, then $j^+(k)\in\mathcal{E}_{\lambda}$ as well. We therefore define a binary operation $*$ on $\mathcal{E}_{\lambda}$ called application defined by $j*k=j^{+}(k)$. The binary operation $*$ together with composition $\circ$ satisfies the following identities:<br />
<br />
1. $(j\circ k)\circ l=j\circ(k\circ l),\,j\circ k=(j*k)\circ j,\,j*(k*l)=(j\circ k)*l,\,j*(k\circ l)=(j*k)\circ(j*l)$<br />
<br />
2. $j*(k*l)=(j*k)*(j*l)$ (self-distributivity).<br />
<br />
Identity 2 is an algebraic consequence of the identities in 1.<br />
<br />
If $j\in\mathcal{E}_{\lambda}$ is a nontrivial elementary embedding, then $j$ freely generates a subalgebra of $(\mathcal{E}_{\lambda},*,\circ)$ with respect to the identities in 1, and $j$ freely generates a subalgebra of $(\mathcal{E}_{\lambda},*)$ with respect to the identity 2.<br />
<br />
If $j_{n}\in\mathcal{E}_{\lambda}$ for all $n\in\omega$, then $\sup\{\textrm{crit}(j_{0}*\dots*j_{n})\mid n\in\omega\}=\lambda$ where the implied parentheses a grouped on the left (for example, $j*k*l=(j*k)*l$).<br />
<br />
Suppose now that $\gamma$ is a limit ordinal with $\gamma<\lambda$. Then define an equivalence relation $\equiv^{\gamma}$ on $\mathcal{E}_{\lambda}$ where $j\equiv^{\gamma}k$ if and only if $j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$ for each $x\in V_{\gamma}$. Then the equivalence relation $\equiv^{\gamma}$ is a congruence on the algebra $(\mathcal{E}_{\lambda},*,\circ)$. In other words, if $j_{1},j_{2},k\in \mathcal{E}_{\lambda}$ and $j_{1}\equiv^{\gamma}j_{2}$ then $j_{1}\circ k\equiv^{\gamma} j_{2}\circ k$ and $j_{1}*k\equiv^{\gamma}j_{2}*k$, and if $j,k_{1},k_{2}\in\mathcal{E}_{\lambda}$ and $k_{1}\equiv^{\gamma}k_{2}$ then $j\circ k_{1}\equiv^{\gamma}j\circ k_{2}$ and $j*k_{1}\equiv^{j(\gamma)}j*k_{2}$.<br />
<br />
If $\gamma<\lambda$, then every finitely generated subalgebra of $(\mathcal{E}_{\lambda}/\equiv^{\gamma},*,\circ)$ is finite.<br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Zero_sharp&diff=2074Zero sharp2017-11-11T21:16:06Z<p>Wabb2t: </p>
<hr />
<div>[[Category:Large cardinal axioms]]<br />
[[Category:Constructibility]]<br />
$0^{\#}$ is a [[projective|$\Sigma_3^1$]] real number which cannot be proven to exist in [[ZFC|$\text{ZFC}$]]. It's existence contradicts the [[Axiom of constructibility]], $V=L$. In fact, it's existence is somewhat equivalent to $L$ being completely different from $V$. <br />
<br />
== Definition ==<br />
<br />
$0^{\#}$ is defined as the set of all Gödel numberings of first-order formula $\varphi$ such that $L\models\varphi(\aleph_0,\aleph_1...\aleph_n)$ for some $n$. Because of the [[stable|stability]] of $\aleph_\omega$, $0^{\#}$ is equivalent to the set of all Gödel numberings of first-order formula $\varphi$ such that $L_{\aleph_{\omega}}\models\varphi(\aleph_0,\aleph_1...\aleph_n)$. This definition implies the existence of Silver Indiscernables. Moreover, it implies:<br />
<br />
*Given any set $X\in L$ which is first-order definable in $L$, $X\in L_{\omega_1}$. This of course implies that $\aleph_1$ is not first-order definable in $L$, because $\aleph_1\not\in L_{\omega_1}$. This is already a disproof of $V=L$ (because $\aleph_1$ is first-order definable).<br />
*For every $\alpha\in\omega_1^L$, every uncountable cardinal is [[Ramsey#iterable|$\alpha$-iterable]], $\geq$ an [[Erdos|$\alpha$-Erdős]], and [[ineffable|totally ineffable]] in $L$.<br />
*There are $\mathfrak{c}$ many reals which are not constructible (that is, $x\not\in L$).<br />
<br />
The existence of $0^\#$ is implied by:<br />
* Chang's Conjecture.<br />
* The existence of an $\omega_1$-iterable cardinal.<br />
* The negation of the singular cardinal hypothesis ($\text{SCH}$).<br />
* The [[axiom of determinacy]] ($\text{AD}$).<br />
<br />
== $0^{\#}$ cardinal ==<br />
<br />
$0^{\#}$ exists iff there is a nontrivial [[elementary embedding]] $j:L\rightarrow L$ (by a theorem of Kunen). The critical point of such an embedding is sometimes called a $0^{\#}$ cardinal, and sometimes called a $j:L\rightarrow L$ cardinal. These cardinals do not coincide with measurable cardinals for a long time. While the least measurable cardinal is [[indescribable|$\Sigma_1^2$-describable]], each of these cardinals is totally indescribable. Furthermore, the least measurable cardinal $\kappa$ such that $V_\kappa$ satisfies the existence of a measurable cardinal is not a $j:L\rightarrow L$ cardinal, and the least measurable cardinal $\kappa$ such that $V_\kappa$ satisfies the existence of such a cardinal is not a $j:L\rightarrow L$ cardinal, and so on.<br />
<br />
However, the existence of a measurable suffices to prove the existence and consistency of a $j:L\rightarrow L$ cardinal.<br />
<br />
''More information to be added here.''<br />
<br />
== References ==<br />
*Jech, Thomas J. Set Theory (The 3rd Millennium Ed.). Springer, 2003.</div>Wabb2thttp://cantorsattic.info/index.php?title=Zero_sharp&diff=2073Zero sharp2017-11-11T21:14:17Z<p>Wabb2t: </p>
<hr />
<div>[[Category:Large cardinal axioms]]<br />
[[Category:Constructibility]]<br />
$0^{\#}$ is a [[projective|$\Sigma_3^1$]] real number which cannot be proven to exist in [[ZFC|$\text{ZFC}$]]. It's existence contradicts the [[Axiom of constructibility]], $\text{V=L}$. In fact, it's existence is somewhat equivalent to $\text{L}$ being completely different from $\text{V}$. <br />
<br />
== Definition ==<br />
<br />
$0^{\#}$ is defined as the set of all Gödel numberings of first-order formula $\varphi$ such that $\text{L}\models\varphi(\aleph_0,\aleph_1...\aleph_n)$ for some $n$. Because of the [[stable|stability]] of $\aleph_\omega$, $0^{\#}$ is equivalent to the set of all Gödel numberings of first-order formula $\varphi$ such that $\text{L}_{\aleph_{\omega}}\models\varphi(\aleph_0,\aleph_1...\aleph_n)$. This definition implies the existence of Silver Indiscernables. Moreover, it implies:<br />
<br />
*Given any set $X\in\text{L}$ which is first-order definable in $\text{L}$, $X\in\text{L}_{\omega_1}$. This of course implies that $\aleph_1$ is not first-order definable in $\text{L}$, because $\aleph_1\not\in\text{L}_{\omega_1}$. This is already a disproof of $\text{V=L}$ (because $\aleph_1$ is first-order definable).<br />
*For every $\alpha\in\omega_1^\text{L}$, every uncountable cardinal is [[Ramsey#iterable|$\alpha$-iterable]], $\geq$ an [[Erdos|$\alpha$-Erdős]], and [[ineffable|totally ineffable]] in $\text{L}$.<br />
*There are $\mathfrak{c}$ many reals which are not constructible (that is, $x\not\in\text{L}$).<br />
<br />
The existence of $0^\#$ is implied by:<br />
* Chang's Conjecture.<br />
* The existence of an $\omega_1$-iterable cardinal.<br />
* The negation of the singular cardinal hypothesis ($\text{SCH}$).<br />
* The [[axiom of determinacy]] ($\text{AD}$).<br />
<br />
== $0^{\#}$ cardinal ==<br />
<br />
$0^{\#}$ exists iff there is a nontrivial [[elementary embedding]] $j:\text{L}\rightarrow\text{L}$ (by a theorem of Kunen). The critical point of such an embedding is sometimes called a $0^{\#}$ cardinal, and sometimes called a $j:\text{L}\rightarrow\text{L}$ cardinal. These cardinals do not coincide with measurable cardinals for a long time. While the least measurable cardinal is [[indescribable|$\Sigma_1^2$-describable]], each of these cardinals is totally indescribable. Furthermore, the least measurable cardinal $\kappa$ such that $V_\kappa$ satisfies the existence of a measurable cardinal is not a $j:\text{L}\rightarrow\text{L}$ cardinal, and the least measurable cardinal $\kappa$ such that $V_\kappa$ satisfies the existence of such a cardinal is not a $j:\text{L}\rightarrow\text{L}$ cardinal, and so on.<br />
<br />
However, the existence of a measurable suffices to prove the existence and consistency of a $j:\text{L}\rightarrow\text{L}$ cardinal.<br />
<br />
''More information to be added here.''<br />
<br />
== References ==<br />
*Jech, Thomas J. Set Theory (The 3rd Millennium Ed.). Springer, 2003.</div>Wabb2thttp://cantorsattic.info/index.php?title=Upper_attic&diff=2072Upper attic2017-11-11T21:10:51Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE:The upper attic}}<br />
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]<br />
[[Category:Large cardinal axioms]]<br />
<br />
Welcome to the upper attic, the transfinite realm of large cardinals, the higher infinite, carrying us upward from the merely inaccessible and indescribable to the subtle and endlessly extendible concepts beyond, towards the calamity of inconsistency. <br />
<br />
* The '''[[Kunen inconsistency]]''': [[Reinhardt]] cardinal, [[Kunen_inconsistency#Super_Reinhardt_cardinal | super Reinhardt]] cardinal, [[Berkeley]] cardinal<br />
* '''[[Rank into rank]]''' cardinals $j:V_\lambda\to V_\lambda$, [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$, I0 cardinal [[L of V_lambda+1 | $j:\text{L}(V_{\lambda+1})\to\text{L}(V_{\lambda+1})$]]<br />
* The [[wholeness axioms]]<br />
* [[n-fold variants | n-fold supercompact]], [[n-fold variants | n-fold strong]], [[n-fold variants | n-fold extendible]], [[n-fold variants | n-fold Woodin]]<br />
* '''[[huge|n-huge]]''' cardinal, [[huge | almost n-huge]] cardinal, [[huge|super n-huge]] cardinal, [[superstrong|(n+1)-superstrong cardinal]]<br />
* [[high-jump]] cardinal, [[high-jump|almost high-jump]] cardinal, [[high-jump|super high-jump]] cardinal, [[high-jump|high-jump with unbounded excess closure]] cardinal<br />
* [[Vopenka | Vopěnka's principle]], [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin|Woodin for supercompactness]] cardinal<br />
* [[extendible]] cardinal, [[extendible | $\alpha$-extendible]] cardinal<br />
<!--* [[grand reflection]] cardinal--><br />
* [[hypercompact]] cardinal<br />
* '''[[supercompact]]''' cardinal, [[supercompact | $\lambda$-supercompact]] cardinal, [[PFA|$\text{PFA}$]] cardinal<br />
* '''[[strongly compact]]''' cardinal<br />
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals<br />
* [[Weakly_compact#Indestructibility of a weakly compact cardinal|indestructible weakly compact]] cardinal<br />
* [[subcompact]] cardinal<br />
* [[superstrong]] cardinal<br />
* [[Shelah]] cardinal<br />
* The '''[[axiom of determinacy]]''' and [[axiom of projective determinacy|its projective counterpart]]<br />
* '''[[Woodin]]''' cardinal<br />
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchies<br />
* [[tall]] cardinal<br />
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br />
* [[zero dagger| $0^\dagger$]], $j:\text{L}[U]\to\text{L}[U]$ cardinal<br />
* '''[[measurable]]''' cardinal, [[weakly measurable]] cardinal<br />
* '''[[Ramsey]]''' cardinal, [[strongly Ramsey]] cardinal, [[virtually Ramsey]] cardinal<br />
* [[Rowbottom]] cardinal<br />
* [[Jonsson | Jónsson]] cardinal<br />
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$ <br />
* '''[[zero sharp | $0^\sharp$]]''', $j:\text{L}\to\text{L}$ cardinal<br />
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br />
* the [[Ramsey#.24.5Calpha.24-iterable cardinal| $\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$<br />
* [[remarkable]] cardinal<br />
* [[ineffable]] cardinal, [[weakly ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy; [[completely ineffable]] cardinal<br />
* [[subtle]] cardinal<br />
* [[ineffable#Ethereal cardinal|ethereal]] cardinal<br />
* [[unfoldable#superstrongly unfoldable cardinal | superstrongly unfoldable]] cardinal, [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br />
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br />
* [[unfoldable]] cardinal, [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal<br />
* [[indescribable]] cardinal, [[totally indescribable]] cardinal<br />
* '''[[weakly compact]]''' cardinal<br />
* The [[Positive set theory|positive set theory]] $\text{GPK}^+_\infty$ <br />
* '''[[Mahlo]]''' cardinal, [[Mahlo#Hyper-Mahlo | $1$-Mahlo]], the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy, [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals<br />
* [[uplifting]] cardinal, [[uplifting#pseudo uplifting cardinal | pseudo uplifting]] cardinal<br />
* [[ORD is Mahlo|$\text{Ord}$ is Mahlo]]<br />
* [[reflecting#Sigma_2 correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals<br />
* [[inaccessible#Degrees of inaccessibility | $1$-inaccessible]], the [[inaccessible#Degrees of inaccessibility | $\alpha$-inaccessible]] hierarchy and [[inaccessible#Hyper-inaccessible | hyper-inaccessible]] cardinals<br />
* [[inaccessible#Universes | Grothendieck universe axiom]], equivalent to the existence of a proper class of [[inaccessible]] cardinals<br />
* '''[[inaccessible]]''' cardinal, '''[[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]]''' cardinal<br />
* [[Morse-Kelley set theory|Morse-Kelley]] set theory<br />
* '''[[worldly]]''' cardinal and the [[worldly#Degrees of worldliness | $\alpha$-wordly]] hierarchy, [[worldly#Degrees of worldliness | hyper-worldly]] cardinal<br />
* the [[Transitive ZFC model#Transitive model universe axiom | transitive model universe axiom]] <br />
* [[transitive ZFC model|transitive model of $\text{ZFC}$]]<br />
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]]<br />
* '''[[Con ZFC | $\text{Con(ZFC)}$]]''' and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]]<br />
* '''[[ZFC|Zermelo-Fraenkel]]''' set theory<br />
<br />
* down to [[the middle attic]]</div>Wabb2thttp://cantorsattic.info/index.php?title=Woodin&diff=2071Woodin2017-11-11T21:08:02Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE:Woodin cardinal}}<br />
'''Woodin cardinals''' (named after W. Hugh Woodin) are a generalization of the notion of strong cardinals and have been used to calibrate the exact proof-theoretic strength of the [[axiom of determinacy]]. They can also be seen as weakenings of ''Shelah cardinals'', defined below. Their exact definition has several equivalent but different characterizations, each of which is somewhat technical in nature. Nevertheless, an inner model theory encapsulating infinitely many Woodin cardinals and slightly beyond has been developed.<br />
<br />
== Definition and some properties ==<br />
<br />
We first introduce the concept of ''$\gamma$-strongness for $A$'': an ordinal $\kappa$ is ''$\gamma$-strong for $A$'' (or $\gamma$-$A$-strong) if there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\kappa$ such that $V_{\kappa+\gamma}\subseteq M$ and $A\cap V_{\kappa+\gamma} = j(A)\cap V_{\kappa+\gamma}$. Intuitively, $j$ preserves $A$.<br />
<br />
We also introduce ''Woodin-ness in $\delta$'': for an infinite ordinal $\delta$, a set $X\subseteq\delta$ is ''Woodin in $\delta$'' if for every function $f:\delta\to\delta$, there is an ordinal $\kappa\in X$ with $\{f(\beta)|\beta<\kappa\}\subseteq\kappa$, there exists a nontrivial [[elementary embedding]] $j:V\to M$ with critical point $\kappa$ such that $V_{j(f)(\kappa)}\subseteq M$.<br />
<br />
An [[inaccessible]] cardinal $\delta$ is '''Woodin''' if any of the following (equivalent) characterizations holds <cite>Kanamori2009:HigherInfinite</cite>:<br />
* For any set $A\subseteq V_\delta$, there exists a $\kappa<\delta$ that is $\gamma$-strong for $A$ for every $\gamma<\kappa$.<br />
* For any set $A\subseteq V_\delta$, the set $S=\{\kappa<\delta|\kappa$ is $\gamma$-strong for $A$ for every $\gamma<\kappa\}$ is [[stationary]] in $\delta$.<br />
* The set $F=\{X\subseteq \delta|\delta\setminus X$ is not ''Woodin in $\delta$''$\}$ is a proper [[filter]], the ''Woodin filter'' over $\delta$.<br />
* For every function $f:\delta\to\delta$ there exists $\kappa<\delta$ such that $\{f(\beta)|\beta\in\kappa\}\subseteq\kappa$ and there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\kappa$ such that $V_{j(f)(\delta)}\subseteq M$.<br />
<br />
Let $\delta$ be Woodin, $F$ be the Woodin filter over $\delta$, and $S=\{\kappa<\delta|\kappa$ is $\gamma$-strong for $A$ for every $\gamma<\kappa\}$. Then $F$ is normal and $S\in F$. <cite>Kanamori2009:HigherInfinite</cite> This implies every Woodin cardinal is [[Mahlo]] and preceeded by a stationary set of [[measurable]] cardinals. However, Woodin cardinals are not [[weakly compact]] as they are ''not'' $\Pi^1_1$-[[indescribable]].<br />
<br />
Woodin cardinals are weaker consistency-wise then [[superstrong]] cardinals. In fact, every superstrong is preceeded by a stationary set of Woodin cardinals.<br />
<br />
The existence of a Woodin cardinal implies the consistency of $\text{ZFC}$ + "the [[filter|nonstationary ideal]] on $\omega_1$ is $\aleph_2$-saturared". [[Huge]] cardinals were first invented to prove the consistency of the existence of a $\aleph_2$-saturated ideal on $\omega_1$, but turned out to be stronger than required, as a Woodin is enough.<br />
<br />
== Shelah cardinals ==<br />
<br />
Shelah cardinals were introduced by Shelah and Woodin as a weakening of the necessary hypothesis required to show several regularity properties of sets of reals hold in the model $\text{L}(\mathbb{R})$ (e.g., every set of reals is Lebesgue measurable and has the property of Baire, etc...). In slightly more detail, Woodin had established that the [[axiom of determinacy]] (a hypothesis known to imply regularity properties for sets of reals) holds in $\text{L}(\mathbb{R})$ <!--(see [[constructible universe]])-->assuming the existence of a nontrivial elementary embedding $j:\text{L}(V_{\lambda+1})\to \text{L}(V_{\lambda+1})$ with critical point $<\lambda$. This axiom, a [[rank-into-rank]] axiom, is known to be very strong and its use was first weakened to that of the existence of a [[supercompact]] cardinal. Following the work of Foreman, Magidor and Shelah on saturated ideals on $\omega_1$, Woodin and Shelah subsequently isolated the two large cardinal hypotheses which bear their name and turn out to be sufficient to establish the [[projective#Regularity properties|regularity properties]] of sets of reals mentioned above.<br />
<br />
Shelah cardinals were the first cardinals to be devised by Woodin and Shelah. A cardinal $\delta$ is ''Shelah'' if for every function $f:\delta\to\delta$ there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\delta$ such that $V_{j(f)(\delta)}\subseteq M$. Every Shelah is Woodin, but not every Woodin is Shelah: indeed, Shelah cardinals are always measurable and in fact [[strong]], while Woodins are usually not. However, just like Woodins, Shelah cardinals are weaker consistency-wise than superstrong cardinals.<br />
<br />
A related notion is ''Shelah-for-supercompactness'', where the closure condition $V_{j(f)(\delta)}\subseteq M$ is replaced by $M^{j(f)(\delta)}\subseteq M$, a much stronger condition. The difference between Shelah and Shelah-for-supercompactness cardinals is essentially the same as the difference between strong and [[supercompact]] cardinals, or between [[superstrong]] and [[huge]] cardinals. Also, just like every Shelah is preceeded by a stationary set of strong cardinals, every Shelah-for-supercompactness cardinal is preceeded by a stationary set of supercompact cardinals. ''Woodin-for-supercompactness'' cardinals were also considered, but they turned out to be equivalent to [[Vopenka|Vopěnka]] cardinals.<br />
<br />
== Woodin cardinals and determinacy ==<br />
<br />
''See also: [[axiom of determinacy]], [[projective#Projective determinacy|axiom of projective determinacy]]''<br />
<br />
Woodin cardinals are linked to different forms of the [[axiom of determinacy]] <cite>Kanamori2009:HigherInfinite</cite><cite>Larson2010:HistoryDeterminacy</cite><cite>KoellnerWoodin2010:LCFD</cite>:<br />
* $\text{ZF+AD}$, $\text{ZFC+AD}^{\text{L}(\mathbb{R})}$, ZFC+"the non-stationary ideal over $\omega_1$ is $\omega_1$-dense" and $\text{ZFC}$+"there exists infinitely many Woodin cardinals" are equiconsistent.<br />
* Under $\text{ZF+AD}$, the model $\text{HOD}^{\text{L}(\mathbb{R})}$ satisfies $\text{ZFC}$+"$\Theta^{\text{L}(\mathbb{R})}$ is a Woodin cardinal". <cite>KoellnerWoodin2010:LCFD</cite> gives many generalization of this result.<br />
* If there exists infinitely many Woodin cardinals with a measurable above them all, then $\text{AD}^{\text{L}(\mathbb{R})}$. If there assumtion that there is a measurable above those Woodins is removed, one still has projective determinacy.<br />
* In fact projective determinacy is equivalent to "for every $n<\omega$, there is a fine-structural, countably iterable inner model $M$ such that $M$ satisfies $\text{ZFC}$+"there exists $n$ Woodin cardinals".<br />
* For every $n$, if there exists $n$ Woodin cardinals with a measurable above them all, then all $\mathbf{\Sigma}^1_{n+1}$ sets are determined.<br />
* $\mathbf{\Pi}^1_2$-determinacy is equivalent to "for every $x\in\mathbb{R}$, there is a countable ordinal $\delta$ such that $\delta$ is a Woodin cardinal in some inner model of $\text{ZFC}$ containing $x$.<br />
* $\mathbf{\Delta}^1_2$-determinacy is equivalent to "for every $x\in\mathbb{R}$, there is an inner model M such that $x\in M$ and $M$ satisfies ZFC+"there is a Woodin cardinal". <br />
* $\text{ZFC}$ + ''lightface'' $\Delta^1_2$-determinacy implies that there many $x$ such that $\text{HOD}^{\text{L}[x]}$ satisfies $\text{ZFC}$+"$\omega_2^{L[x]}$ is a Woodin cardinal".<br />
* $\text{Z}_2+\Delta^1_2$-determinacy is conjectured to be equiconsistent with $\text{ZFC}$+"$\text{Ord}$ is Woodin", where "$\text{Ord}$ is Woodin" is expressed as an axiom scheme and $\text{Z}_2$ is [[:wikipedia:second-order arithmetic|second-order arithmetic]].<br />
* $\text{Z}_3+\Delta^1_2$-determinacy is provably equiconsistent with $\text{NBG}$+"$\text{Ord}$ is Woodin" where $\text{NBG}$ is [[:wikipedia:Von Neumann–Bernays–Gödel set theory|Von Neumann–Bernays–Gödel set theory]] and $\text{Z}_3$ is third-order arithmetic.<br />
<br />
== Role in $\Omega$-logic ==<br />
== Stationary tower forcing ==<br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Kunen_inconsistency&diff=2070Kunen inconsistency2017-11-11T21:01:24Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE:The Kunen inconsistency}}<br />
<br />
The Kunen inconsistency, the theorem showing that there can be no nontrivial [[elementary embedding]] from the universe to itself, remains a focal point of large cardinal set theory, marking a hard upper bound at the summit of the main ascent of the large cardinal hierarchy, the first outright refutation of a large cardinal axiom. On this main ascent, large cardinal axioms assert the existence of elementary embeddings $j:V\to M$ where $M$ exhibits increasing affinity with $V$ as one climbs the hierarchy. The $\theta$-[[strong]] cardinals, for example, have $V_\theta\subset M$; the $\lambda$-[[supercompact]] cardinals have $M^\lambda\subset M$; and the [[huge]] cardinals have $M^{j(\kappa)}\subset M$. The natural limit of this trend, first suggested by Reinhardt, is a nontrivial elementary embedding $j:V\to V$, the critical point of which is accordingly known as a ''Reinhardt''<br />
cardinal. Shortly after this idea was introduced, however, Kunen famously proved that there are no such embeddings, and hence no Reinhardt cardinals in $\text{ZFC}$. <br />
<br />
Since that time, the inconsistency argument has been generalized by various authors, including Harada<br />
<cite>Kanamori2009:HigherInfinite</cite>(p. 320-321),<br />
Hamkins, Kirmayer and Perlmutter <cite>HamkinsKirmayerPerlmutter:GeneralizationsOfKunenInconsistency</cite>, Woodin <cite>Kanamori2009:HigherInfinite</cite>(p. 320-321),<br />
Zapletal <cite>Zapletal1996:ANewProofOfKunenInconsistency</cite> and Suzuki <cite>Suzuki1998:NojVtoVinVofG, Suzuki1999:NoDefinablejVtoVinZF</cite>.<br />
<br />
* There is no nontrivial elementary embedding $j:V\to V$ from the set-theoretic universe to itself.<br />
* There is no nontrivial elementary embedding $j:V[G]\to V$ of a set-forcing extension of the universe to the universe, and neither is there $j:V\to V[G]$ in the converse direction.<br />
* More generally, there is no nontrivial elementary embedding between two ground models of the universe.<br />
* More generally still, there is no nontrivial elementary embedding $j:M\to N$ when both $M$ and $N$ are eventually stationary correct.<br />
* There is no nontrivial elementary embedding $j:V\to \text{HOD}$, and neither is there $j:V\to M$ for a variety of other definable classes, including $\text{gHOD}$ and the $\text{HOD}^\eta$, $\text{gHOD}^\eta$.<br />
* If $j:V\to M$ is elementary, then $V=\text{HOD}(M)$.<br />
* There is no nontrivial elementary embedding $j:\text{HOD}\to V$.<br />
* More generally, for any definable class $M$, there is no nontrivial elementary embedding $j:M\to V$.<br />
* There is no nontrivial elementary embedding $j:\text{HOD}\to\text{HOD}$ that is definable in $V$ from parameters.<br />
<br />
It is not currently known whether the Kunen inconsistency may be undertaken in ZF. Nor is it known whether one may rule out nontrivial embeddings $j:\text{HOD}\to\text{HOD}$ even in $\text{ZFC}$.<br />
<br />
== Metamathematical issues ==<br />
<br />
Kunen formalized his theorem in Kelly-Morse set theory, but it is also possble to prove it in the weaker system of G&ouml;del-Bernays set theory. In each case, the embedding $j$ is a $\text{GBC}$ class, and elementary of $j$ is asserted as a $\Sigma_1$-elementary embedding, which implies $\Sigma_n$-elementarity when the two models have the ordinals.<br />
<br />
== Reinhardt cardinal ==<br />
<br />
Although the existence of Reinhardt cardinals has now been refuted in $\text{ZFC}$ and $\text{GBC}$, the term is used in the $\text{ZF}$ context to refer to the critical point of a nontrivial elementary embedding $j:V\to V$ of the set-theoretic universe to itself.<br />
<br />
== Super Reinhardt cardinal ==<br />
<br />
A ''super Reinhardt'' cardinal $\kappa$, is a cardinal which is the critical point of elementary embeddings $j:V\to V$, with $j(\kappa)$ as large as desired.<br />
<br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Con_ZFC&diff=2069Con ZFC2017-11-11T20:59:44Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: $\text{Con(ZFC)}$}}<br />
The assertion $\text{Con(ZFC)}$ is the assertion that the theory $\text{ZFC}$ is consistent. This is an assertion with complexity $\Pi^0_1$ in arithmetic, since it is the assertion that every natural number is not the G&ouml;del code of the proof of a contradiction from $\text{ZFC}$. Because of the G&ouml;del completeness theorem, the assertion is equivalent to the assertion that the theory $\text{ZFC}$ has a model $\langle M,\hat\in\rangle$. One such model is the Henkin model, built in the syntactic procedure from any complete consistent Henkin theory extending $\text{ZFC}$. In general, one may not assume that $\hat\in$ is the actual set membership relation, since this would make the model a [[transitive ZFC model | transitive model of $\text{ZFC}$]], whose existence is a strictly stronger assertion than $\text{Con(ZFC)}$. <br />
<br />
The G&ouml;del incompleteness theorem implies that if $\text{ZFC}$ is consistent, then it does not prove $\text{Con(ZFC)}$, and so the addition of this axiom is strictly stronger than $\text{ZFC}$ alone. <br />
<br />
== Consistency hierarchy == <br />
<br />
The expression $\text{Con}^2(\text{ZFC})$ denotes the assertion $\text{Con}(\text{ZFC}+\text{Con}(\text{ZFC}))$, and iterating this more generally, one may consider the assertion $\text{Con}^\alpha(\text{ZFC})$ whenever $\alpha$ itself is expressible.<br />
<br />
== Every model of $\text{ZFC}$ contains a model of $\text{ZFC}$ as an element ==<br />
<br />
Every model $M$ of $\text{ZFC}$ has an element $N$, which it believes to be a first-order structure in the language of set theory, which is a model of $\text{ZFC}$, as viewed externally from $M$. This is clear in the case where $M$ is an [[omega model | $\omega$-model]] of $\text{ZFC}$, since in this case $M$ agrees that $\text{ZFC}$ is consistent and can therefore build a Henkin model of $\text{ZFC}$. In the remaining case, $M$ has nonstandard natural numbers. By the [[reflection theorem]] applied in $M$, we know that the $\Sigma_n$ fragment of $\text{ZFC}$ is true in models of the form $V_\beta^M$, for every standard natural number $n$. Since $M$ cannot identify its standard cut, it follows that there must be some nonstandard $n$ for which $M$ thinks some $V_\beta^M$ satisfies the (nonstandard) $\Sigma_n$ fragment of $\text{ZFC}$. Since $n$ is nonstandard, this includes the full standard theory of $\text{ZFC}$, as desired. <br />
<br />
The fact mentioned in the previous paragraph is occasionally found to be surprising by some beginning set-theorists, perhaps because naively the conclusion seems to contradict the fact that there can be models of $\text{ZFC}+\neg\text{Con}(\text{ZFC})$. The paradox is resolved, however, by realizing that although the model $N$ inside $M$ is actually a model of full $\text{ZFC}$, the model $M$ need not agree that it is a model of $\text{ZFC}$, in the case that $M$ has nonstandard natural numbers and hence nonstandard length axioms of $\text{ZFC}$.</div>Wabb2thttp://cantorsattic.info/index.php?title=Worldly&diff=2068Worldly2017-11-11T20:57:58Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: Worldly cardinal}}<br />
A cardinal $\kappa$ is ''worldly'' if $V_\kappa$ is a model of $\text{ZFC}$. It follows that $\kappa$ is a [[strong limit]], a [[beth fixed point]] and a fixed point of the enumeration of these, and more.<br />
<br />
* Every [[inaccessible]] cardinal is worldly.<br />
* Nevertheless, the least worldly cardinal is [[singular]] and hence not [[inaccessible]]. <br />
* The least worldly cardinal has [[cofinality]] $\omega$.<br />
* Indeed, the next worldly cardinal above any ordinal, if any exist, has [[cofinality]] $\omega$. <br />
* Any worldly cardinal $\kappa$ of uncountable cofinality is a limit of $\kappa$ many worldly cardinals.<br />
<br />
==Degrees of worldliness==<br />
<br />
A cardinal $\kappa$ is ''$1$-worldly'' if it is worldly and a limit of worldly cardinals. More generally, $\kappa$ is ''$\alpha$-worldly'' if it is worldly and for every $\beta\lt\alpha$, the $\beta$-worldly cardinals are unbounded in $\kappa$. The cardinal $\kappa$ is ''hyper-worldly'' if it is $\kappa$-worldly. One may proceed to define notions of $\alpha$-hyper-worldly and $\alpha$-hyper${}^\beta$-worldly in analogy with the [[inaccessible#hyper-inaccessible | hyper-inaccessible cardinals]]. Every [[inaccessible]] cardinal $\kappa$ is hyper${}^\kappa$-worldly, and a limit of such kinds of cardinals.<br />
<br />
The worldly cardinal terminology was introduced in lectures of J. D. Hamkins at the CUNY Graduate Center and at NYU.</div>Wabb2thttp://cantorsattic.info/index.php?title=Transitive_ZFC_model&diff=2067Transitive ZFC model2017-11-11T20:57:11Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: Transitive model of $\text{ZFC}$}}<br />
A ''transitive model of $\text{ZFC}$'' is a [[transitive]] set $M$ such that the structure $\langle M,\in\rangle$ satisfies all of the [[ZFC|$\text{ZFC}$]] axioms of set theory. The existence of such a model is strictly stronger than [[Con ZFC | $\text{Con(ZFC)}$]] and stronger than an iterated [[Con ZFC#Consistency hierarchy | consistency hierarchy]], but weaker than the existence of an [[worldly]] cardinal, a cardinal $\kappa$ for which $V_\kappa$ is a model of $\text{ZFC}$, and consequently also weaker than the existence of an [[inaccessible]] cardinal. Not all transitive models of $\text{ZFC}$ have the $V_\kappa$ form, for if there is any transitive model of $\text{ZFC}$, then by the L&ouml;wenheim-Skolem theorem there is a countable such model, and these never have the form $V_\kappa$. <br />
<br />
Nevertheless, every transitive model $M$ of $\text{ZFC}$ provides a set-theoretic forum inside of which one can view nearly all classical mathematics taking place. In this sense, such models are inaccessible to or out of reach of ordinary set-theoretic constructions. As a result, the existence of a transitive model of $\text{ZFC}$ can be viewed as a large cardinal axiom: it expresses a notion of largeness, and the existence of such a model is not provable in $\text{ZFC}$ and has consistency strength strictly exceeding $\text{ZFC}$. <br />
<br />
== Minimal transitive model of $\text{ZFC}$ ==<br />
<br />
If there is any transitive model $M$ of $\text{ZFC}$, then $L^M$, the constructible universe as computed in $M$, is also a transitive model of $\text{ZFC}$ and indeed, has the form $L_\eta$, where $\eta=\text{ht}(M)$ is the height of $M$. The ''minimal transitive model of $\text{ZFC}$'' is the model $L_\eta$, where $\eta$ is smallest such that this is a model of $\text{ZFC}$. The argument just given shows that the minimal transitive model is a subset of all other transitive models of $\text{ZFC}$. <br />
<br />
== Omega model of $\text{ZFC}$ == <br />
<br />
An ''$\omega$-model'' of $\text{ZFC}$ is a model of $\text{ZFC}$ whose collection of natural numbers is isomorphic to the actual natural numbers. In other words, an $\omega$-model is a model having no nonstandard natural numbers, although it may have nonstandard ordinals. (More generally, for any ordinal $\alpha$, an $\alpha$-model has well-founded part at least $\alpha$.) Every transitive model of $\text{ZFC}$ is an $\omega$-model, but the latter concept is strictly weaker.<br />
<br />
== Consistency hierarchy ==<br />
<br />
The existence of an $\omega$-model of $\text{ZFC}$ and implies [[Con ZFC | $\text{Con(ZFC)}$]], of course, and also [[ Con ZFC#Consistency hierarchy | $\text{Con(ZFC+Con(ZFC))}$]] and a large part of the iterated [[ Con ZFC#Consistency hierarchy | consistency hierarchy]]. This is simply because if $M\models\text{ZFC}$ and has the standard natural numbers, then $M$ agrees that $\text{Con(ZFC)}$ holds, since it has the same proofs as we do in the ambient background. Thus, we believe that $M$ satisfies $\text{ZFC+Con(ZFC)}$ and consequently we believe $\text{Con(ZFC+Con(ZFC))}$. It follows again that $M$ agrees with this consistency assertion, and so we now believe $\text{Con}^3(\text{ZFC})$. The model $M$ therefore agrees and so we believe $\text{Con}^4(\text{ZFC})$ and so on transfinitely, as long as we are able to describe the ordinal iterates in a way that $M$ interprets them correctly.<br />
<br />
== Transitive models of $\text{ZFC}$ fragments ==<br />
<br />
Every finite fragment of $\text{ZFC}$ admits numerous transitive models, as a consequence of the [[reflection theorem]]. <br />
<br />
== Transitive models and forcing ==<br />
<br />
Countable transitive models of set theory were used historically as a convenient way to formalize [[forcing]]. Such models $M$ make the theory of forcing convenient, since one can easily prove that for every partial order $\mathbb{P}$ in $M$, there is an $M$-generic [[filter]] $G\subset\mathbb{P}$, simply by enumerating the dense subsets of $\mathbb{P}$ in $M$ in a countable sequence $\langle D_n\mid n\lt\omega\rangle$, and building a descending sequence $p_0\geq p_1\geq p_2\geq\cdots$, with $p_n\in D_n$. The filter $G$ generated by the sequence is $M$-generic. <br />
<br />
For the purposes of consistency proofs, this manner of formalization worked quite well. To show $\text{Con}(\text{ZFC})\to \text{Con}(\text{ZFC}+\varphi)$, one fixes a finite fragment of $\text{ZFC}$ and works with a countable transitive model of a suitably large fragment, producing $\varphi$ with the desired fragment in a forcing extension of it.<br />
<br />
== Transitive model universe axiom == <br />
<br />
The ''transitive model universe axiom'' is the assertion that every set is an element of a transitive model of $\text{ZFC}$. This axiom makes a stronger claim than the [[reflecting#The Feferman theory | Feferman theory]], since it is asserted as a single first-order claim, but weaker than the [[universe axiom]], which asserts that the universes have the form $V_\kappa$ for inaccessible cardinals $\kappa$. <br />
<br />
The transitive model universe axiom is sometimes studied in the background theory not of $\text{ZFC}$, but of [[ZFC-P]], omitting the power set axiom, together with the axiom asserting that every set is countable. Such an enterprise amounts to adopting the latter theory, not as the fundamental axioms of mathematics, but rather as a background meta-theory for studying the [[multiverse]] perspective, investigating how the various actual set-theoretic universe, transitive models of full $\text{ZFC}$, relate to one another.</div>Wabb2thttp://cantorsattic.info/index.php?title=Axiom_of_determinacy&diff=2066Axiom of determinacy2017-11-11T20:52:40Z<p>Wabb2t: </p>
<hr />
<div>The '''axiom of determinacy''' is the assertion that a certain type of two-player games of perfect information (i.e. games in which the players alternate moves which are known to both players, and the outcome of the game depends only on this list of moves, and not on chance or other external factors) are ''determined'', that is, there is an "optimal strategy" that allows one player to win regardless of the other player's strategy. That strategy is called a ''winning strategy'' for that player.<br />
<br />
The axiom of determinacy is incompatible with the [[axiom of choice]]. More precisely, it is incompatible with the existence of a well-ordering of the reals. The $\text{AD}$ is, however, not known to be inconsistent with [[ZFC|$\text{ZF}$]] set theory. $\text{AD}$ is furthermore a very powerful axiom, as $\text{ZF+AD}$ implies the consistency of $\text{ZF}$, $\text{ZF+Con(ZFC)}$, and much more - it is in fact close of being a large cardinal axiom, as Woodin proved that it was equiconsistent with the existence of infinitely many [[Woodin]] cardinals. <cite>KoellnerWoodin2010:LCFD</cite><br />
<br />
It follows from large cardinal axioms (in particular from the existence of infinitely many Woodins with a [[measurable]] above them all <cite>KoellnerWoodin2010:LCFD</cite>) that the $\text{AD}$ is true in $\text{L}(\mathbb{R})$, the [[constructible universe]] obtained by starting with the transitive closure of the set of all reals (i.e. $\text{L}_0(\mathbb{R})=\text{TC}(\{\mathbb{R}\})$). This assertion, generally refered to as $\text{L}(\mathbb{R})$-determinacy, $\text{AD}^{\text{L}(\mathbb{R})}$ <cite>KoellnerWoodin2010:LCFD</cite> or ''quasi-projective determinacy'' ($\text{QPD}$) <cite>Maddy88:BelAxiomsII</cite> is not known to be inconsistent with $\text{ZFC}$. $\text{AD}^{\text{L}(\mathbb{R})}$ is furthermore equiconsistent with $\text{AD}$ (in $\text{V}$). A particular case of this is the [[axiom of projective determinacy]] ($\text{PD}$) which states that every [[projective]] set is determined, projectivity being a weak form of definability (more precisely definability in second-order arithmethic).<br />
<br />
== Type of games that are determined ==<br />
<br />
Given a set $S$ of infinite sequences of order-type (length) $\omega$ (i.e, a subset of the Baire Space $\omega^{\omega}$), the ''payoff'' set, the game begins as such: Player I says a natural number $n_0$, then Player II says a natural number $n_1$, and so on, until a sequence of order-type $\omega$ is constructed. At this point, a natural number $n_i$ has been given for every natural number $i$. Player I wins if $(n_0,n_1,n_2...)\in S$, Player II wins otherwise. Since $\omega^\omega$ and the set $\mathbb{R}$ of the real numbers are in bijection with the other, we shall often identify the elements of $\omega^\omega$ as the ''real numbers'', like if $\omega^\omega$ and $\mathbb{R}$ were equal. Thus the game considered here produces a real number.<br />
<br />
A ''strategy'' for player I (resp. player II) is a function $\Sigma$ with domain the set of sequences of integers of even (odd) length such that for each $a\in dom(\Sigma)$, $\Sigma(a)\in\omega$. A run of the game (partial or complete) is said to be according to a strategy $\Sigma$ for player player I (player II) if every initial segment of the run of odd (nonzero even) length is of the form $a\frown⟨\Sigma(a)⟩$ for some sequence $a$. A strategy $\Sigma$ for player player I (player II) is a winning strategy if every complete run of the game according to $\Sigma$ is in (out of) $S$. We say that a set $S\subset\omega^\omega$ is determined if there exists a winning strategy for one of the players<br />
<br />
The '''axiom of determinacy''' ($\text{AD}$) states that every payoff set $S\subset\omega^\omega$ is determined <cite>Larson2010:HistoryDeterminacy</cite>. It is possible to show that every finite or countable payoff set is determined, so this equivalent to the assertion that every uncountable payoff set is determined.<br />
<br />
== Refuting the axiom of determinacy from a well-ordering of the reals ==<br />
<br />
As stated above, the axiom of determinacy is not compatible with the axiom of choice, that is, within $\text{ZFC}$ we can prove that axiom of determinacy fails. We outline a construction of an undetermined game starting from a well-ordering of continuum.<br />
<br />
A strategy for either player is a function with countable domain (a subset of the set of all finite sequences of integers) to $\omega$, so there are $2^{\aleph_0}$ many strategies for player I and $2^{\aleph_0}$ continuum many strategies for player II. Let $\{s^{I}_\alpha:\alpha<2^{\aleph_0}\}, \{s^{II}_\alpha:\alpha<2^{\aleph_0}\}$ be enumerations of strategies for the respective players. We shall now construct, by transfinite recursion, two disjoint sets of sequences $\{a_\alpha:\alpha<2^{\aleph_0}\}, \{b_\alpha:\alpha<2^{\aleph_0}\}\subseteq\omega^\omega$ such that $\{a_\alpha:\alpha<2^{\aleph_0}\}$ is not determined.<br />
<br />
Suppose that, for some $\beta<2^{\aleph_0}$, $\{a_\alpha:\alpha<\beta\},\{b_\alpha:\alpha<\beta\}$ have already been constructed. Take strategy $s^I_\beta$. There are continuum many possible plays according to this strategy (since player II can play in arbitrary way at any of their turns), so not all of them can be already contained in $\{a_\alpha:\alpha<\beta\}$ (which has cardinality $|\beta|<2^{\aleph_0}$). Therefore, using well-ordering of continuum, we can pick one of these plays and define it to be $b_\beta$. Similarly, we can pick $a_\beta$ according to strategy $s^{II}_\beta$ which is not already in $\{b_\alpha:\alpha\leq\beta\}$. This way the sets $\{a_\alpha:\alpha<2^{\aleph_0}\},\{b_\alpha:\alpha<2^{\aleph_0}\}$ are clearly disjoint.<br />
<br />
Letting $A=\{a_\alpha:\alpha<2^{\aleph_0}\}$, we now claim the game with payoff set $A$ is undetermined. Indeed, suppose player I has a winning strategy. This must be one of the strategies $s^I_\beta$. By construction, player II can arrange the play so that the resulting play is $b_\beta$ (since we have chosen it so that it's consistent with strategy $b_\beta$), which is not an element of $A$, contradicting the assumption that $s^I_\beta$ is a winning strategy. Analogously, for any strategy $s^{II}_\beta$ for player II, player I can force the play to be $a_\beta\in A$. Therefore no strategy for either player is a winning strategy and it follows that the game is undetermined.<br />
<br />
== Other known limitations of determinacy ==<br />
<br />
Assuming the axiom of choice there is a non-determined game of length $\omega$. However, choice isn't known to contradict the determinacy of all ''definable'' games of length $\omega$.<br />
<br />
With or without assuming choice, there is a non-determined game of length $\omega_1$ and a a non-determined definable game of length $\omega_1+\omega$. There is also a non-determined game of length $\omega$ with moves in $\omega_1$ (i.e. the payoff sets are subsets of $\omega_1^\omega$ instead of subsets of $\omega^\omega$. There is a non-determined game of length $\omega$ with moves in $\mathcal{P}(\mathbb{R})$, and using choice one can show there is such a game that is definable. [http://mathoverflow.net/questions/271507/limitations-of-determinacy-hypotheses-in-zfc]<br />
<br />
Definable games of length $\omega$ with moves in $\mathbb{R}$ are provably determined from large cardinal axioms. Determinacy of such games that are projective follows from the existence of sufficiently many Woodin cardinals.<br />
<br />
By a result of Woodin, if there is an iterable model of $\text{ZFC}$ with a countable (in $\text{V}$) Woodin cardinal which is a limit of Woodin cardinals, then it is consistent (even with choice) that all ordinal-definable games of length $\omega_1$ are determined. This is only a consistency result, not a proof of "all ordinal-definable games of length $\omega_1$ are determined".<br />
<br />
== Implications of the axiom of determinacy ==<br />
<br />
Assume $\text{ZF+AD}$. Most of the following results can be found in <cite>Kanamori2009:HigherInfinite</cite>, in <cite>Larson2010:HistoryDeterminacy</cite> or in [http://mathoverflow.net/questions/129036/counterintuitive-consequences-of-the-axiom-of-determinacy]:<br />
<br />
* The [[:wikipedia:axiom of countable choice|axiom of countable choice]] restrained to countable sets of reals is true.<br />
* In $\text{L}(\mathbb{R})$ the [[:wikipedia:axiom of dependent choice|axiom of dependent choice]] is true.<br />
* The reals cannot be well-ordered. Thus the full [[axiom of choice]] fails.<br />
* Every set of reals is [[:wikipedia:Lebesgue measurable|Lebesgue measurable]]. Thus the [[:wikipedia:Banach-Tarski paradox|Banach-Tarski paradox]] fails.<br />
** It follows by a theorem of Raisonnier that $\omega_1\not\leq 2^{\aleph_0}$ (yet $\omega_1\not\geq2^{\aleph_0}$).<br />
** Furthermore, it implies $2^{\aleph_0}$ can be partitioned in more than $2^{\aleph_0}$ many pairwise disjoint nonempty subsets.<br />
* Every set of reals has the [[:wikipedia:Baire property|Baire property]].<br />
* Every set of reals is either [[countable]] or has a [[:wikipedia:perfect set|perfect subset]].<br />
** Thus a form of the [[continuum hypothesis]] holds, i.e. every set of reals is either countable or has cardinality $2^{\aleph_0}$.<br />
** Other forms of $\text{CH}$ however fail, in particular $2^{\aleph_0}\neq\aleph_1$.<br />
* There are no free [[filter|ultrafilters]] on $\omega$. Every ultrafilter on $\omega$ is principal. Thus every ultrafilter is countably complete ($\aleph_1$-complete).<br />
* $\omega_1$, $\omega_2$, and $\omega_{\omega+1}$ and $\omega_{\omega+2}$ are all [[measurable]] cardinals.<br />
** The [[club]] filter on $\omega_1$ is an ultrafilter. Every subset of $\omega_1$ either contains a club or is disjoint from one.<br />
** The club filter on $\omega_2$ restrained to sets of [[cofinality]] $\omega$ is $\omega_2$-complete.<br />
* $\omega_n$ is singular for every $n>2$ and has cofinality $\omega_2$ and is [[Jonsson]], also $\aleph_\omega$ is [[Rowbottom]].<br />
* [[zero sharp|$0^{\#}$]] exists, thus the [[:wikipedia:axiom of constructibility|axiom of constructibility]] ($\text{V=L}$) fails.<br />
** In fact, $x^{\#}$ exists for every $x\in\mathbb{R}$ (thus $\text{V}\neq\text{L}[x]$).<br />
* The strong [[partition property]], $\omega_1\rightarrow(\omega_1)^{\omega_1}_2$, holds. In fact, $\omega_1\rightarrow(\omega_1)^{\omega_1}_{2^{\aleph_0}}$ and $\omega_1\rightarrow(\omega_1)^{\omega_1}_\alpha$ for every $\alpha<\omega_1$.<br />
* If there is a surjection $\mathbb{R}\to\alpha$, then there is surjection $\mathbb{R}\to\mathcal{P}(\alpha)$ (this is ''Moschovakis' coding lemma'').<br />
* [[:wikipedia:Hall's marriage theorem|Hall's marriage theorem]] fails for infinite graphs. For example there is there is a 2-regular bipartite graph on $\mathbb{R}$ with no perfect matching.<br />
* There is no [[:wikipedia:Basis (linear algebra)#Related_notions|Hamel basis]] of $\mathbb{R}$ over $\mathbb{Q}$.<br />
<br />
Let $\Theta$ be the supremum of the ordinals that $\mathbb{R}$ can be mapped onto. Under $\text{AC}$ this is just $(2^{\aleph_0})^{+}$ but under $\text{AD}$ it is a limit cardinal, in fact an aleph fixed point, and $\text{DC}$ implies it has uncountable cofinality. In $\text{L}(\mathbb{R})$ it is also regular and thus [[weakly inaccessible]]. It is conjectured that under $\text{AD}$ the cofinality function is nondecreasing on singular cardinals below $\Theta$.<br />
<br />
== Determinacy of $\text{L}(\mathbb{R})$ ==<br />
<br />
''See also: [[Constructible universe]]''<br />
<br />
Recall that a formula $\varphi$ is $\Delta_0$ if and only if it only contains bounded quantifiers (i.e. $(\forall x\in y)$ and $(\exists x\in y)$). Let $\text{def}(X)=\{Y\subset X : Y$ is first-order definable by a $\Delta_0$ formula with parameters only from $X\cup\{X\}\}$. Then let:<br />
*$\text{L}_0(X)=\text{TC}(\{X\})$<br />
*$\text{L}_{\alpha+1}(X)=\text{def}(\text{L}_\alpha(X))$<br />
*$\text{L}_\lambda(X)=\bigcup_{\alpha<\lambda}\text{L}_\alpha(X)$ for limit $\lambda$<br />
*$\text{L}(X)=\bigcup_{\alpha\in \text{Ord}}\text{L}_\alpha(X)$<br />
where $\text{TC}({X})$ is the smallest transitive set containing $X$, the elements of $X$, the elements of the elements of $X$, and so on. $\text{L}(X)$is always a model of $\text{ZF}$, but not necessarily of the axiom of choice.<br />
<br />
$\text{L}(X,Y)$ is used as a shortcut for $\text{L}(\{X,Y\})$. $\text{L}(X,\mathbb{R})$ with $X\subset\mathbb{R}$ is different from $\text{L}(\mathbb{R})$ whenever $X$ is not constructible from the reals, i.e. $X\not\in \text{L}(\mathbb{R})$ (if any such set exists; it is consistent with $\text{ZF+AD}$ that they do not). <br />
<br />
$\text{L}(\mathbb{R})$-determinacy, also known as $\text{AD}^{\text{L}(\mathbb{R})}$ <cite>KoellnerWoodin2010:LCFD</cite> or ''quasi-projective determinacy'' <cite>Maddy88:BelAxiomsII</cite> is the assertion that every set of reals in $\text{L}(\mathbb{R})$ is determined. Equivalently, "$\text{L}(\mathbb{R})$ is a model of $\text{ZF+AD}$".<br />
<br />
$\text{AD}^{\text{L}(\mathbb{R})}$ appears to be a very "natural" statement in that, empirically, every natural extension of $\text{ZFC}$ (i.e. not made specifically to contradict this) that is not proved consistent by $\text{AD}$ seems to imply $\text{AD}^{\text{L}(\mathbb{R})}$ or some weaker form of determinacy. <cite>Larson2010:HistoryDeterminacy</cite> This is often considered to be an argument toward the "truth" of $\text{AD}^{\text{L}(\mathbb{R})}$.<br />
<br />
Assuming $\text{ZF+DC+V=L(}\mathbb{R})$, $\text{AD}$ follows from three of its consequences: <cite>Larson2010:HistoryDeterminacy</cite><br />
# Every set of reals is Lebesgue measurable.<br />
# Every set of reals has the Baire property. <br />
# Every $\Sigma^1_2$ set of reals can be uniformized.<br />
<br />
In $\text{L}(\mathbb{R})$, the axiom of determinacy is equivalent to the axiom of Turing determinacy <cite>Larson2010:HistoryDeterminacy</cite>, i.e. the assertion that payoff sets closed under [[:wikipedia:Turing equivalence|Turing equivalence]] are determined.<br />
<br />
Busche and Schindler showed that, if there is a model of $\text{ZF}$ in wich every uncountable cardinal is singular (thus has cofinality $\aleph_0$), then the axiom of determinacy holds in the $\text{L}(\mathbb{R})$ of some forcing extension of $\text{HOD}$ <cite>Larson2010:HistoryDeterminacy</cite>. This notably follows from the existence of a proper class of [[strongly compact]] cardinals. <br />
<br />
Assume that there is $\omega_1$-dense ideal over $\omega_1$; then $\text{AD}^{\text{L}(\mathbb{R})}$ holds. <cite>Kanamori2009:HigherInfinite</cite> This result is due to Woodin.<br />
<br />
The following holds in $\text{L}(\mathbb{R})$ assuming $\text{AD}^{\text{L}(\mathbb{R})}$: <cite>KoellnerWoodin2010:LCFD</cite><cite>JacksonKetchersidSchlutzenbergWoodin:DeterminacyJonsson</cite> <br />
* Every uncountable cardinal $<\Theta$ is [[Jonsson|Jónsson]], also if it is regular or has cofinality $\omega$ then it is [[Rowbottom]].<br />
* Every regular cardinal $<\Theta$ is [[measurable]] (note that $2^{\aleph_0}\not\leq\Theta$), also $\Theta$ is a limit of measurable cardinals.<br />
* $\Theta$ is weakly $\Theta$-[[Mahlo]] (and thus weakly $\Theta$-inaccessible), but it is not [[weakly compact]].<br />
* $\omega_1$ is <$\Theta$-[[supercompact]], i.e. it is $\gamma$-supercompact for all $\gamma<\Theta$.<br />
* $\Theta$ is Woodin in the model $\text{HOD}^{\text{L}(\mathbb{R})}$.<br />
<br />
== Axiom of projective determinacy ==<br />
<br />
''Main article: [[Projective#Projective determinacy|Projective determinacy]]''<br />
<br />
== Axiom of real determinacy ==<br />
<br />
The '''axiom of real determinacy''' ($\text{AD}_\mathbb{R}$) is the assertion that if payoff sets contains real numbers instead of natural numbers, then every payoff set is still determined. This is strictly stronger than AD, and $\text{ZF+AD}_\mathbb{R}$ proves $\text{ZF+AD}$ consistent.<br />
<br />
$\text{AD}_\mathbb{R}$ is equivalent (over $\text{ZF}$) to $\text{AD}$ plus the [[:wikipedia:Uniformization (set theory)|axiom of uniformization]] (which is false in $\text{L}(\mathbb{R})$). $\text{AD}_\mathbb{R}$ is also equivalent to determinacy for games of length $\omega^2$. In fact, $\text{AD}_\mathbb{R}$ is equivalent to the assertion that every game of bounded countable length is determined. It is however possible to show (in $\text{ZF}$) that there are non-determined games of length $\aleph_1$.<br />
<br />
Solovay showed that $\text{ZF+AD}_\mathbb{R}$+"$\Theta$ has uncountable cofinality" (which follows from $\text{ZF+AD}_\mathbb{R}\text{+DC}$) proves $\text{ZF+AD}_\mathbb{R}$ consistent; it is therefore consistent with $\text{ZF+AD}_\mathbb{R}$ that $\Theta$ has cofinality $\omega$ and that $\text{DC}$ is false. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
Steel showed that under $\text{AD}_\mathbb{R}$, in a forcing extension there is a proper class model of $\text{ZFC}$ in which there exists a cardinal $\delta$ of cofinality $\aleph_0$ which is a limit of Woodin cardinals and <$\delta$-strong cardinals. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
Under $\text{AD}_\mathbb{R}$, $\omega_1$ is <$\Theta$-supercompact, i.e. for every ordinal $\gamma<\Theta$ there is a normal fine ultrafilter on the set of all subsets of $\gamma$ of size $\aleph_1$. $\text{AD}$ suffices for this result to hold in $\text{L}(\mathbb{R})$, but is not known to suffice for it to hold in $\text{V}$. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
A set $\Gamma\subset\mathcal{P}(\mathbb{R})$ is a ''Wadge initial segment'' of $\mathcal{P}(\mathbb{R})$ if for every $X\in\Gamma$, if $Y\leq_W X$ (i.e. $Y$ is [[:wikipedia:Wadge hierarchy|Wadge reducible]] to $X$) then $Y\in\Gamma$. Under suitable large cardinal assumptions, there exists a Wadge initial segment $\Gamma\subset\mathcal{P}(\mathbb{R})$ such that $L(\Gamma,\mathbb{R})\models\text{AD}^{+}+\text{AD}_\mathbb{R}+\Gamma=\mathcal{P}(\mathbb{R})$ (see [[:wikipedia:AD+|AD+]]). Furthermore, whenever $\mathcal{M}$ is an inner model such that $\mathbb{R}\subset\mathcal{M}$ and $\mathcal{M}\models\text{AD}^{+}+\text{AD}_\mathbb{R}$, one has $\Gamma\subset\mathcal{M}$. ''(see the 'Read more' section)''<br />
<br />
== Consistency strength of determinacy hypotheses ==<br />
<br />
The following theories are equiconsistent: <cite>Kanamori2009:HigherInfinite</cite><cite>TrangWilson2016:DetFromStrongCompactness</cite><br />
* $\text{ZF+AD}$<br />
* $\text{ZF+AD+DC}$<br />
* $\text{ZFC+AD}^{\text{L}(\mathbb{R})}$<br />
* $\text{ZFC+AD}^{\text{OD}(\mathbb{R})}$<br />
* $\text{ZFC+}$"the non-stationary ideal over $\omega_1$ is $\omega_1$-dense"<br />
* $\text{ZFC+}$"there exists infinitely many [[Woodin]] cardinals"<br />
* $\text{ZF+DC+}$"$\omega_1$ is $\mathcal{P}(\omega_1)$-[[strongly compact]]"<br />
* $\text{ZF+DC+}$"$\omega_1$ is $\mathbb{R}$-strongly compact and $\Theta>\omega_2$"<br />
* $\text{ZF+DC+}$"$\omega_1$ is $\mathbb{R}$-strongly compact and $\omega_2$-strongly compact"<br />
* $\text{ZF+DC+}$"$\omega_1$ is $\mathbb{R}$-strongly compact and Jensens's square principle fails for $\omega_1$"<br />
Where $\text{DC}$ is the [[:wikipedia:axiom of dependent choice|axiom of dependent choice]] and $\omega_1$ being $X$-strongly compact means that there exists a [[filter|fine measure]] on the set of all subsets of $X$ of cardinality $\aleph_1$.<br />
<br />
[[Projective determinacy]] is a little weaker: it is equiconsistent with $\text{ZFC}$ plus, for all n, an axiom saying "there are n Woodin cardinals". Since $\text{ZFC}$ can only use finitely many of its axioms, this axiom schema does not allow $\text{ZFC}$ to prove that there exists infinitely many Woodins, despite making it able to prove every particular instance of "there exists at least n Woodin cardinals".<br />
<br />
Koellner annd Woodin showed that the following theories are also equiconsistent: <cite>KoellnerWoodin2010:LCFD</cite><br />
* $\text{ZFC+}\Delta^1_2$-determinacy<br />
* $\text{ZFC+OD}$-determinacy<br />
* $\text{ZFC+}$"there exists a Woodin cardinal"<br />
<br />
And so are $\text{Z}_3$+lightface $\Delta^1_2$-determinacy and $\text{MK+}$"$\text{Ord}$ is Woodin" where $\text{Z}_3$ is ''third-order arithmetic'' and $\text{MK}$ is [[Morse-Kelley set theory]]. It is also conjectured that $\text{Z}_2+\Delta^1_2$-determinacy and $\text{ZFC+}$"$\text{Ord}$ is Woodin" are equiconsistent, where $\text{Z}_2$ is [[:wikipedia:second-order arithmetic|second-order arithmetic]] and "$\text{Ord}$ is Woodin" is expressed as an axiom scheme.<br />
<br />
Finally, Trang and Wilson proved that the following theories are equiconsistent: <cite>TrangWilson2016:DetFromStrongCompactness</cite><br />
* $\text{ZF+DC+AD}_\mathbb{R}$<br />
* $\text{ZF+DC+}$"$\omega_1$ is $\mathcal{P}(\mathbb{R})$-strongly compact"<br />
* $\text{ZF+DC+}$"$\omega_1$ is $\mathbb{R}$-strongly compact and $\Theta$ is singular"<br />
* $\text{ZF+DC+}$"$\omega_1$ is $\mathbb{R}$-strongly compact and $\Theta$-strongly compact"<br />
As are the following theories:<br />
* $\text{ZF+AD}_\mathbb{R}$<br />
* $\text{ZF+DC}_{\mathcal{P}(\omega_1)}$+"$\omega_1$ is $\mathbb{R}$-strongly compact and $\Theta$ is singular"<br />
<br />
== Read more ==<br />
<br />
* ''"Is there a natural inner model of $\text{AD}_\mathbb{R}$?"'' [http://mathoverflow.net/questions/269241/is-there-a-natural-inner-model-of-ad-mathbbr/269690]<br />
<br />
* ''"Limitations of determinacy hypotheses in ZFC"'' [http://mathoverflow.net/questions/271507/limitations-of-determinacy-hypotheses-in-zfc]<br />
<br />
* ''"Counterintuitive consequences of the Axiom of Determinacy?"'' [https://mathoverflow.net/questions/129036/counterintuitive-consequences-of-the-axiom-of-determinacy]<br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Axiom_of_determinacy&diff=2065Axiom of determinacy2017-11-11T20:51:47Z<p>Wabb2t: </p>
<hr />
<div>The '''axiom of determinacy''' is the assertion that a certain type of two-player games of perfect information (i.e. games in which the players alternate moves which are known to both players, and the outcome of the game depends only on this list of moves, and not on chance or other external factors) are ''determined'', that is, there is an "optimal strategy" that allows one player to win regardless of the other player's strategy. That strategy is called a ''winning strategy'' for that player.<br />
<br />
The axiom of determinacy is incompatible with the [[axiom of choice]]. More precisely, it is incompatible with the existence of a well-ordering of the reals. The $\text{AD}$ is, however, not known to be inconsistent with [[ZFC|$\text{ZF}$]] set theory. $\text{AD}$ is furthermore a very powerful axiom, as $\text{ZF+AD}$ implies the consistency of $\text{ZF}$, $\text{ZF+Con(ZFC)}$, and much more - it is in fact close of being a large cardinal axiom, as Woodin proved that it was equiconsistent with the existence of infinitely many [[Woodin]] cardinals. <cite>KoellnerWoodin2010:LCFD</cite><br />
<br />
It follows from large cardinal axioms (in particular from the existence of infinitely many Woodins with a [[measurable]] above them all <cite>KoellnerWoodin2010:LCFD</cite>) that the $\text{AD}$ is true in $\text{L}(\mathbb{R})$, the [[constructible universe]] obtained by starting with the transitive closure of the set of all reals (i.e. $\text{L}_0(\mathbb{R})=\text{TC}(\{\mathbb{R}\})$). This assertion, generally refered to as $\text{L}(\mathbb{R})$-determinacy, $\text{AD}^{\text{L}(\mathbb{R})}$ <cite>KoellnerWoodin2010:LCFD</cite> or ''quasi-projective determinacy'' ($\text{QPD}$) <cite>Maddy88:BelAxiomsII</cite> is not known to be inconsistent with $\text{ZFC}$. $\text{AD}^{\text{L}(\mathbb{R})}$ is furthermore equiconsistent with $\text{AD}$ (in $V$). A particular case of this is the [[axiom of projective determinacy]] ($\text{PD}$) which states that every [[projective]] set is determined, projectivity being a weak form of definability (more precisely definability in second-order arithmethic).<br />
<br />
== Type of games that are determined ==<br />
<br />
Given a set $S$ of infinite sequences of order-type (length) $\omega$ (i.e, a subset of the Baire Space $\omega^{\omega}$), the ''payoff'' set, the game begins as such: Player I says a natural number $n_0$, then Player II says a natural number $n_1$, and so on, until a sequence of order-type $\omega$ is constructed. At this point, a natural number $n_i$ has been given for every natural number $i$. Player I wins if $(n_0,n_1,n_2...)\in S$, Player II wins otherwise. Since $\omega^\omega$ and the set $\mathbb{R}$ of the real numbers are in bijection with the other, we shall often identify the elements of $\omega^\omega$ as the ''real numbers'', like if $\omega^\omega$ and $\mathbb{R}$ were equal. Thus the game considered here produces a real number.<br />
<br />
A ''strategy'' for player I (resp. player II) is a function $\Sigma$ with domain the set of sequences of integers of even (odd) length such that for each $a\in dom(\Sigma)$, $\Sigma(a)\in\omega$. A run of the game (partial or complete) is said to be according to a strategy $\Sigma$ for player player I (player II) if every initial segment of the run of odd (nonzero even) length is of the form $a\frown⟨\Sigma(a)⟩$ for some sequence $a$. A strategy $\Sigma$ for player player I (player II) is a winning strategy if every complete run of the game according to $\Sigma$ is in (out of) $S$. We say that a set $S\subset\omega^\omega$ is determined if there exists a winning strategy for one of the players<br />
<br />
The '''axiom of determinacy''' ($\text{AD}$) states that every payoff set $S\subset\omega^\omega$ is determined <cite>Larson2010:HistoryDeterminacy</cite>. It is possible to show that every finite or countable payoff set is determined, so this equivalent to the assertion that every uncountable payoff set is determined.<br />
<br />
== Refuting the axiom of determinacy from a well-ordering of the reals ==<br />
<br />
As stated above, the axiom of determinacy is not compatible with the axiom of choice, that is, within $\text{ZFC}$ we can prove that axiom of determinacy fails. We outline a construction of an undetermined game starting from a well-ordering of continuum.<br />
<br />
A strategy for either player is a function with countable domain (a subset of the set of all finite sequences of integers) to $\omega$, so there are $2^{\aleph_0}$ many strategies for player I and $2^{\aleph_0}$ continuum many strategies for player II. Let $\{s^{I}_\alpha:\alpha<2^{\aleph_0}\}, \{s^{II}_\alpha:\alpha<2^{\aleph_0}\}$ be enumerations of strategies for the respective players. We shall now construct, by transfinite recursion, two disjoint sets of sequences $\{a_\alpha:\alpha<2^{\aleph_0}\}, \{b_\alpha:\alpha<2^{\aleph_0}\}\subseteq\omega^\omega$ such that $\{a_\alpha:\alpha<2^{\aleph_0}\}$ is not determined.<br />
<br />
Suppose that, for some $\beta<2^{\aleph_0}$, $\{a_\alpha:\alpha<\beta\},\{b_\alpha:\alpha<\beta\}$ have already been constructed. Take strategy $s^I_\beta$. There are continuum many possible plays according to this strategy (since player II can play in arbitrary way at any of their turns), so not all of them can be already contained in $\{a_\alpha:\alpha<\beta\}$ (which has cardinality $|\beta|<2^{\aleph_0}$). Therefore, using well-ordering of continuum, we can pick one of these plays and define it to be $b_\beta$. Similarly, we can pick $a_\beta$ according to strategy $s^{II}_\beta$ which is not already in $\{b_\alpha:\alpha\leq\beta\}$. This way the sets $\{a_\alpha:\alpha<2^{\aleph_0}\},\{b_\alpha:\alpha<2^{\aleph_0}\}$ are clearly disjoint.<br />
<br />
Letting $A=\{a_\alpha:\alpha<2^{\aleph_0}\}$, we now claim the game with payoff set $A$ is undetermined. Indeed, suppose player I has a winning strategy. This must be one of the strategies $s^I_\beta$. By construction, player II can arrange the play so that the resulting play is $b_\beta$ (since we have chosen it so that it's consistent with strategy $b_\beta$), which is not an element of $A$, contradicting the assumption that $s^I_\beta$ is a winning strategy. Analogously, for any strategy $s^{II}_\beta$ for player II, player I can force the play to be $a_\beta\in A$. Therefore no strategy for either player is a winning strategy and it follows that the game is undetermined.<br />
<br />
== Other known limitations of determinacy ==<br />
<br />
Assuming the axiom of choice there is a non-determined game of length $\omega$. However, choice isn't known to contradict the determinacy of all ''definable'' games of length $\omega$.<br />
<br />
With or without assuming choice, there is a non-determined game of length $\omega_1$ and a a non-determined definable game of length $\omega_1+\omega$. There is also a non-determined game of length $\omega$ with moves in $\omega_1$ (i.e. the payoff sets are subsets of $\omega_1^\omega$ instead of subsets of $\omega^\omega$. There is a non-determined game of length $\omega$ with moves in $\mathcal{P}(\mathbb{R})$, and using choice one can show there is such a game that is definable. [http://mathoverflow.net/questions/271507/limitations-of-determinacy-hypotheses-in-zfc]<br />
<br />
Definable games of length $\omega$ with moves in $\mathbb{R}$ are provably determined from large cardinal axioms. Determinacy of such games that are projective follows from the existence of sufficiently many Woodin cardinals.<br />
<br />
By a result of Woodin, if there is an iterable model of $\text{ZFC}$ with a countable (in $V$) Woodin cardinal which is a limit of Woodin cardinals, then it is consistent (even with choice) that all ordinal-definable games of length $\omega_1$ are determined. This is only a consistency result, not a proof of "all ordinal-definable games of length $\omega_1$ are determined".<br />
<br />
== Implications of the axiom of determinacy ==<br />
<br />
Assume $\text{ZF+AD}$. Most of the following results can be found in <cite>Kanamori2009:HigherInfinite</cite>, in <cite>Larson2010:HistoryDeterminacy</cite> or in [http://mathoverflow.net/questions/129036/counterintuitive-consequences-of-the-axiom-of-determinacy]:<br />
<br />
* The [[:wikipedia:axiom of countable choice|axiom of countable choice]] restrained to countable sets of reals is true.<br />
* In $\text{L}(\mathbb{R})$ the [[:wikipedia:axiom of dependent choice|axiom of dependent choice]] is true.<br />
* The reals cannot be well-ordered. Thus the full [[axiom of choice]] fails.<br />
* Every set of reals is [[:wikipedia:Lebesgue measurable|Lebesgue measurable]]. Thus the [[:wikipedia:Banach-Tarski paradox|Banach-Tarski paradox]] fails.<br />
** It follows by a theorem of Raisonnier that $\omega_1\not\leq 2^{\aleph_0}$ (yet $\omega_1\not\geq2^{\aleph_0}$).<br />
** Furthermore, it implies $2^{\aleph_0}$ can be partitioned in more than $2^{\aleph_0}$ many pairwise disjoint nonempty subsets.<br />
* Every set of reals has the [[:wikipedia:Baire property|Baire property]].<br />
* Every set of reals is either [[countable]] or has a [[:wikipedia:perfect set|perfect subset]].<br />
** Thus a form of the [[continuum hypothesis]] holds, i.e. every set of reals is either countable or has cardinality $2^{\aleph_0}$.<br />
** Other forms of $\text{CH}$ however fail, in particular $2^{\aleph_0}\neq\aleph_1$.<br />
* There are no free [[filter|ultrafilters]] on $\omega$. Every ultrafilter on $\omega$ is principal. Thus every ultrafilter is countably complete ($\aleph_1$-complete).<br />
* $\omega_1$, $\omega_2$, and $\omega_{\omega+1}$ and $\omega_{\omega+2}$ are all [[measurable]] cardinals.<br />
** The [[club]] filter on $\omega_1$ is an ultrafilter. Every subset of $\omega_1$ either contains a club or is disjoint from one.<br />
** The club filter on $\omega_2$ restrained to sets of [[cofinality]] $\omega$ is $\omega_2$-complete.<br />
* $\omega_n$ is singular for every $n>2$ and has cofinality $\omega_2$ and is [[Jonsson]], also $\aleph_\omega$ is [[Rowbottom]].<br />
* [[zero sharp|$0^{\#}$]] exists, thus the [[:wikipedia:axiom of constructibility|axiom of constructibility]] ($\text{V=L}$) fails.<br />
** In fact, $x^{\#}$ exists for every $x\in\mathbb{R}$ (thus $\text{V}\neq\text{L}[x]$).<br />
* The strong [[partition property]], $\omega_1\rightarrow(\omega_1)^{\omega_1}_2$, holds. In fact, $\omega_1\rightarrow(\omega_1)^{\omega_1}_{2^{\aleph_0}}$ and $\omega_1\rightarrow(\omega_1)^{\omega_1}_\alpha$ for every $\alpha<\omega_1$.<br />
* If there is a surjection $\mathbb{R}\to\alpha$, then there is surjection $\mathbb{R}\to\mathcal{P}(\alpha)$ (this is ''Moschovakis' coding lemma'').<br />
* [[:wikipedia:Hall's marriage theorem|Hall's marriage theorem]] fails for infinite graphs. For example there is there is a 2-regular bipartite graph on $\mathbb{R}$ with no perfect matching.<br />
* There is no [[:wikipedia:Basis (linear algebra)#Related_notions|Hamel basis]] of $\mathbb{R}$ over $\mathbb{Q}$.<br />
<br />
Let $\Theta$ be the supremum of the ordinals that $\mathbb{R}$ can be mapped onto. Under $\text{AC}$ this is just $(2^{\aleph_0})^{+}$ but under $\text{AD}$ it is a limit cardinal, in fact an aleph fixed point, and $\text{DC}$ implies it has uncountable cofinality. In $\text{L}(\mathbb{R})$ it is also regular and thus [[weakly inaccessible]]. It is conjectured that under $\text{AD}$ the cofinality function is nondecreasing on singular cardinals below $\Theta$.<br />
<br />
== Determinacy of $\text{L}(\mathbb{R})$ ==<br />
<br />
''See also: [[Constructible universe]]''<br />
<br />
Recall that a formula $\varphi$ is $\Delta_0$ if and only if it only contains bounded quantifiers (i.e. $(\forall x\in y)$ and $(\exists x\in y)$). Let $\text{def}(X)=\{Y\subset X : Y$ is first-order definable by a $\Delta_0$ formula with parameters only from $X\cup\{X\}\}$. Then let:<br />
*$\text{L}_0(X)=\text{TC}(\{X\})$<br />
*$\text{L}_{\alpha+1}(X)=\text{def}(\text{L}_\alpha(X))$<br />
*$\text{L}_\lambda(X)=\bigcup_{\alpha<\lambda}\text{L}_\alpha(X)$ for limit $\lambda$<br />
*$\text{L}(X)=\bigcup_{\alpha\in \text{Ord}}\text{L}_\alpha(X)$<br />
where $\text{TC}({X})$ is the smallest transitive set containing $X$, the elements of $X$, the elements of the elements of $X$, and so on. $\text{L}(X)$is always a model of $\text{ZF}$, but not necessarily of the axiom of choice.<br />
<br />
$\text{L}(X,Y)$ is used as a shortcut for $\text{L}(\{X,Y\})$. $\text{L}(X,\mathbb{R})$ with $X\subset\mathbb{R}$ is different from $\text{L}(\mathbb{R})$ whenever $X$ is not constructible from the reals, i.e. $X\not\in \text{L}(\mathbb{R})$ (if any such set exists; it is consistent with $\text{ZF+AD}$ that they do not). <br />
<br />
$\text{L}(\mathbb{R})$-determinacy, also known as $\text{AD}^{\text{L}(\mathbb{R})}$ <cite>KoellnerWoodin2010:LCFD</cite> or ''quasi-projective determinacy'' <cite>Maddy88:BelAxiomsII</cite> is the assertion that every set of reals in $\text{L}(\mathbb{R})$ is determined. Equivalently, "$\text{L}(\mathbb{R})$ is a model of $\text{ZF+AD}$".<br />
<br />
$\text{AD}^{\text{L}(\mathbb{R})}$ appears to be a very "natural" statement in that, empirically, every natural extension of $\text{ZFC}$ (i.e. not made specifically to contradict this) that is not proved consistent by $\text{AD}$ seems to imply $\text{AD}^{\text{L}(\mathbb{R})}$ or some weaker form of determinacy. <cite>Larson2010:HistoryDeterminacy</cite> This is often considered to be an argument toward the "truth" of $\text{AD}^{\text{L}(\mathbb{R})}$.<br />
<br />
Assuming $\text{ZF+DC+V=L(}\mathbb{R})$, $\text{AD}$ follows from three of its consequences: <cite>Larson2010:HistoryDeterminacy</cite><br />
# Every set of reals is Lebesgue measurable.<br />
# Every set of reals has the Baire property. <br />
# Every $\Sigma^1_2$ set of reals can be uniformized.<br />
<br />
In $\text{L}(\mathbb{R})$, the axiom of determinacy is equivalent to the axiom of Turing determinacy <cite>Larson2010:HistoryDeterminacy</cite>, i.e. the assertion that payoff sets closed under [[:wikipedia:Turing equivalence|Turing equivalence]] are determined.<br />
<br />
Busche and Schindler showed that, if there is a model of $\text{ZF}$ in wich every uncountable cardinal is singular (thus has cofinality $\aleph_0$), then the axiom of determinacy holds in the $\text{L}(\mathbb{R})$ of some forcing extension of $\text{HOD}$ <cite>Larson2010:HistoryDeterminacy</cite>. This notably follows from the existence of a proper class of [[strongly compact]] cardinals. <br />
<br />
Assume that there is $\omega_1$-dense ideal over $\omega_1$; then $\text{AD}^{\text{L}(\mathbb{R})}$ holds. <cite>Kanamori2009:HigherInfinite</cite> This result is due to Woodin.<br />
<br />
The following holds in $\text{L}(\mathbb{R})$ assuming $\text{AD}^{\text{L}(\mathbb{R})}$: <cite>KoellnerWoodin2010:LCFD</cite><cite>JacksonKetchersidSchlutzenbergWoodin:DeterminacyJonsson</cite> <br />
* Every uncountable cardinal $<\Theta$ is [[Jonsson|Jónsson]], also if it is regular or has cofinality $\omega$ then it is [[Rowbottom]].<br />
* Every regular cardinal $<\Theta$ is [[measurable]] (note that $2^{\aleph_0}\not\leq\Theta$), also $\Theta$ is a limit of measurable cardinals.<br />
* $\Theta$ is weakly $\Theta$-[[Mahlo]] (and thus weakly $\Theta$-inaccessible), but it is not [[weakly compact]].<br />
* $\omega_1$ is <$\Theta$-[[supercompact]], i.e. it is $\gamma$-supercompact for all $\gamma<\Theta$.<br />
* $\Theta$ is Woodin in the model $\text{HOD}^{\text{L}(\mathbb{R})}$.<br />
<br />
== Axiom of projective determinacy ==<br />
<br />
''Main article: [[Projective#Projective determinacy|Projective determinacy]]''<br />
<br />
== Axiom of real determinacy ==<br />
<br />
The '''axiom of real determinacy''' ($\text{AD}_\mathbb{R}$) is the assertion that if payoff sets contains real numbers instead of natural numbers, then every payoff set is still determined. This is strictly stronger than AD, and $\text{ZF+AD}_\mathbb{R}$ proves $\text{ZF+AD}$ consistent.<br />
<br />
$\text{AD}_\mathbb{R}$ is equivalent (over $\text{ZF}$) to $\text{AD}$ plus the [[:wikipedia:Uniformization (set theory)|axiom of uniformization]] (which is false in $\text{L}(\mathbb{R})$). $\text{AD}_\mathbb{R}$ is also equivalent to determinacy for games of length $\omega^2$. In fact, $\text{AD}_\mathbb{R}$ is equivalent to the assertion that every game of bounded countable length is determined. It is however possible to show (in $\text{ZF}$) that there are non-determined games of length $\aleph_1$.<br />
<br />
Solovay showed that $\text{ZF+AD}_\mathbb{R}$+"$\Theta$ has uncountable cofinality" (which follows from $\text{ZF+AD}_\mathbb{R}\text{+DC}$) proves $\text{ZF+AD}_\mathbb{R}$ consistent; it is therefore consistent with $\text{ZF+AD}_\mathbb{R}$ that $\Theta$ has cofinality $\omega$ and that $\text{DC}$ is false. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
Steel showed that under $\text{AD}_\mathbb{R}$, in a forcing extension there is a proper class model of $\text{ZFC}$ in which there exists a cardinal $\delta$ of cofinality $\aleph_0$ which is a limit of Woodin cardinals and <$\delta$-strong cardinals. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
Under $\text{AD}_\mathbb{R}$, $\omega_1$ is <$\Theta$-supercompact, i.e. for every ordinal $\gamma<\Theta$ there is a normal fine ultrafilter on the set of all subsets of $\gamma$ of size $\aleph_1$. $\text{AD}$ suffices for this result to hold in $\text{L}(\mathbb{R})$, but is not known to suffice for it to hold in $V$. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
A set $\Gamma\subset\mathcal{P}(\mathbb{R})$ is a ''Wadge initial segment'' of $\mathcal{P}(\mathbb{R})$ if for every $X\in\Gamma$, if $Y\leq_W X$ (i.e. $Y$ is [[:wikipedia:Wadge hierarchy|Wadge reducible]] to $X$) then $Y\in\Gamma$. Under suitable large cardinal assumptions, there exists a Wadge initial segment $\Gamma\subset\mathcal{P}(\mathbb{R})$ such that $L(\Gamma,\mathbb{R})\models\text{AD}^{+}+\text{AD}_\mathbb{R}+\Gamma=\mathcal{P}(\mathbb{R})$ (see [[:wikipedia:AD+|AD+]]). Furthermore, whenever $\mathcal{M}$ is an inner model such that $\mathbb{R}\subset\mathcal{M}$ and $\mathcal{M}\models\text{AD}^{+}+\text{AD}_\mathbb{R}$, one has $\Gamma\subset\mathcal{M}$. ''(see the 'Read more' section)''<br />
<br />
== Consistency strength of determinacy hypotheses ==<br />
<br />
The following theories are equiconsistent: <cite>Kanamori2009:HigherInfinite</cite><cite>TrangWilson2016:DetFromStrongCompactness</cite><br />
* $\text{ZF+AD}$<br />
* $\text{ZF+AD+DC}$<br />
* $\text{ZFC+AD}^{\text{L}(\mathbb{R})}$<br />
* $\text{ZFC+AD}^{\text{OD}(\mathbb{R})}$<br />
* $\text{ZFC+}$"the non-stationary ideal over $\omega_1$ is $\omega_1$-dense"<br />
* $\text{ZFC+}$"there exists infinitely many [[Woodin]] cardinals"<br />
* $\text{ZF+DC+}$"$\omega_1$ is $\mathcal{P}(\omega_1)$-[[strongly compact]]"<br />
* $\text{ZF+DC+}$"$\omega_1$ is $\mathbb{R}$-strongly compact and $\Theta>\omega_2$"<br />
* $\text{ZF+DC+}$"$\omega_1$ is $\mathbb{R}$-strongly compact and $\omega_2$-strongly compact"<br />
* $\text{ZF+DC+}$"$\omega_1$ is $\mathbb{R}$-strongly compact and Jensens's square principle fails for $\omega_1$"<br />
Where $\text{DC}$ is the [[:wikipedia:axiom of dependent choice|axiom of dependent choice]] and $\omega_1$ being $X$-strongly compact means that there exists a [[filter|fine measure]] on the set of all subsets of $X$ of cardinality $\aleph_1$.<br />
<br />
[[Projective determinacy]] is a little weaker: it is equiconsistent with $\text{ZFC}$ plus, for all n, an axiom saying "there are n Woodin cardinals". Since $\text{ZFC}$ can only use finitely many of its axioms, this axiom schema does not allow $\text{ZFC}$ to prove that there exists infinitely many Woodins, despite making it able to prove every particular instance of "there exists at least n Woodin cardinals".<br />
<br />
Koellner annd Woodin showed that the following theories are also equiconsistent: <cite>KoellnerWoodin2010:LCFD</cite><br />
* $\text{ZFC+}\Delta^1_2$-determinacy<br />
* $\text{ZFC+OD}$-determinacy<br />
* $\text{ZFC+}$"there exists a Woodin cardinal"<br />
<br />
And so are $\text{Z}_3$+lightface $\Delta^1_2$-determinacy and $\text{MK+}$"$\text{Ord}$ is Woodin" where $\text{Z}_3$ is ''third-order arithmetic'' and $\text{MK}$ is [[Morse-Kelley set theory]]. It is also conjectured that $\text{Z}_2+\Delta^1_2$-determinacy and $\text{ZFC+}$"$\text{Ord}$ is Woodin" are equiconsistent, where $\text{Z}_2$ is [[:wikipedia:second-order arithmetic|second-order arithmetic]] and "$\text{Ord}$ is Woodin" is expressed as an axiom scheme.<br />
<br />
Finally, Trang and Wilson proved that the following theories are equiconsistent: <cite>TrangWilson2016:DetFromStrongCompactness</cite><br />
* $\text{ZF+DC+AD}_\mathbb{R}$<br />
* $\text{ZF+DC+}$"$\omega_1$ is $\mathcal{P}(\mathbb{R})$-strongly compact"<br />
* $\text{ZF+DC+}$"$\omega_1$ is $\mathbb{R}$-strongly compact and $\Theta$ is singular"<br />
* $\text{ZF+DC+}$"$\omega_1$ is $\mathbb{R}$-strongly compact and $\Theta$-strongly compact"<br />
As are the following theories:<br />
* $\text{ZF+AD}_\mathbb{R}$<br />
* $\text{ZF+DC}_{\mathcal{P}(\omega_1)}$+"$\omega_1$ is $\mathbb{R}$-strongly compact and $\Theta$ is singular"<br />
<br />
== Read more ==<br />
<br />
* ''"Is there a natural inner model of $\text{AD}_\mathbb{R}$?"'' [http://mathoverflow.net/questions/269241/is-there-a-natural-inner-model-of-ad-mathbbr/269690]<br />
<br />
* ''"Limitations of determinacy hypotheses in ZFC"'' [http://mathoverflow.net/questions/271507/limitations-of-determinacy-hypotheses-in-zfc]<br />
<br />
* ''"Counterintuitive consequences of the Axiom of Determinacy?"'' [https://mathoverflow.net/questions/129036/counterintuitive-consequences-of-the-axiom-of-determinacy]<br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Projective&diff=2064Projective2017-11-11T20:02:59Z<p>Wabb2t: /* Projective determinacy */</p>
<hr />
<div>{{DISPLAYTITLE: Projective sets and determinacy}}<br />
We say that $\Gamma$ is a ''pointclass'' if it is a collection of subsets of a [[:wikipedia:Polish space|Polish space]]. <br />
The '''lightface and bolface projective hierarchies''' are hierarchies of pointclasses of some Polish space $X$ defined by repeated applications of projections and complementations from either recursively enumerable or closed sets respectively.<br />
<br />
''Most results in this article can be found in <cite>Jech2003:SetTheory</cite> and <cite>Kanamori2009:HigherInfinite</cite> unless indicated otherwise.''<br />
<br />
== Definitions ==<br />
<br />
The following definitions are made by taking $X=\omega^\omega$, the ''Baire space'', i.e. the set of all functions $f:\mathbb{N}\to\mathbb{N}$. We will identify its members with the corresponding real numbers under some fixed bijection between $\mathbb{R}$ and $\omega^\omega$. The definitions presented here can be easily extended to other Polish spaces than the Baire space.<br />
<br />
Let $\mathbf{\Sigma}^0_1$ be the pointclass that contains all open subsets of the Polish space $\omega^\omega$. Let $\mathbf{\Pi}^0_1$ be the pointclass containing the complements of the $\mathbf{\Sigma}^0_1$ sets.<br />
<br />
We define the '''boldface projective pointclasses''' $\mathbf{\Sigma}^1_n$, $\mathbf{\Pi}^1_n$ and $\mathbf{\Delta}^1_n$ the following way:<br />
# $\mathbf{\Sigma}^1_1$ contains all the images of $\mathbf{\Pi}^0_1$ sets by continuous functions; its members are called the ''analytic sets''.<br />
# Now, for all $n$, define $\mathbf{\Pi}^1_n$ to be the set of the complements of the $\mathbf{\Sigma}^1_n$ sets; the members of $\mathbf{\Pi}^1_1$ are called the ''coanalytic sets''.<br />
# For all $n$, define $\mathbf{\Sigma}^1_{n+1}$ to be the set of the images of $\mathbf{\Pi}^1_n$ sets by continuous functions.<br />
# Finally, let $\mathbf{\Delta}^1_n=\mathbf{\Sigma}^1_n\cap\mathbf{\Pi}^1_n$. The members of $\mathbf{\Delta}^1_1$ are the ''Borel'' sets.<br />
<br />
The '''relativized lightface projective pointclasses''' $\Sigma^1_n(a)$, $\Pi^1_n(a)$ and $\Delta^1_n(a)$ (for $a\in\omega^\omega$) are defined similarly except that $\Sigma^1_1(a)$ is defined as the set of all $A\subseteq\omega^\omega$ such that $A=\{x\in\omega^\omega:\exists y\in\omega^\omega$ $\exists n\in\omega$ $R(x\restriction n,y\restriction n,a\restriction n)\}$, that is, $A$ is recursively definable by a formula with only existential quantifiers ranging on members of $\omega^\omega$ or on $\omega$ and whose only parameter is $a$.<br />
<br />
The (non-relativized) '''lightface projective classes''', also known as ''analytical pointclasses'', are the special cases $\Sigma^1_n$, $\Pi^1_n$ and $\Delta^1_n$ of relativized lightface projective pointclasses where $a=\empty$. Let $\Sigma^0_1$ be the pointclass of all ''recursively enumerable'' sets, i.e. the sets $A$ such there exists a recursive relation $R$ such that $A=\{x\in\omega^\omega:\exists n\in\omega$ $R(x\restriction n)\}$, and $\Pi^0_1$ contain the completements of $\Sigma^0_1$ sets. Then the $\Sigma^1_1$ sets are precisely the projections of $\Pi^0_1$ sets.<br />
<br />
Given an arbitrary pointclass $\Gamma$, define $\neg\Gamma$ as the set of the complements of $\Gamma$'s elements, for example $\Pi^1_n(a)=\neg\Sigma^1_n(a)$. Also let $\Delta_\Gamma=\Gamma\cap\neg\Gamma$, for example $\Delta^1_n(a)=\Delta_{\Pi^1_n(a)}=\Delta_{\Sigma^1_n(a)}$.<br />
<br />
== Properties ==<br />
<br />
Every $\mathbf{\Sigma}^1_n$ set is $\Sigma^1_n(a)$ for some $a\in\omega^\omega$, in fact $\mathbf{\Sigma}^1_n=\bigcup_{a\in\omega^\omega}\Sigma^1_n(a)$. A similar statement holds for $\mathbf{\Pi}^1_n$ sets and $\mathbf{\Delta}^1_n$ sets. This means the boldface projective sets are precisely the set definable using only real and arithmetical quantifiers and real parameters.<br />
<br />
The following statements also holds when replacing relativized lightface pointclasses by their boldface counterparts:<br />
* If $A$ and $B$ are $\Sigma^1_n(a)$ relations, then so are $\exists x\in\omega^\omega$ $A$, $A\land B$, $A\lor B$, $\exists n\in\omega$ $A$ and $\forall n\in\omega$ $A$.<br />
* If $A$ and $B$ are $\Pi^1_n(a)$ relations, then so are $\forall x\in\omega^\omega$ $A$, $A\land B$, $A\lor B$, $\exists n\in\omega$ $A$ and $\forall n\in\omega$ $A$.<br />
* If $A$ is a $\Sigma^1_n(a)$ relation then $\neg A$ is a $\Pi^1_n(a)$ relation. If $A$ is $\Pi^1_n(a)$ then $\neg A$ is $\Sigma^1_n(a)$.<br />
* If $A$ is a $\Sigma^1_n(a)$ relation and $B$ is a $\Pi^1_n(a)$ relation, then $A\Rightarrow B$ is a $\Pi^1_n(a)$ relation.<br />
* If $A$ is a $\Pi^1_n(a)$ relation and $B$ is a $\Sigma^1_n(a)$ relation, then $A\Rightarrow B$ is a $\Sigma^1_n(a)$ relation.<br />
* If $A$ and $B$ are $\Delta^1_n(a)$, then so are $\neg A$, $A\land B$, $A\lor B$, $A\Rightarrow B$, $A\Leftrightarrow B$, $\exists n\in\omega$ $A$, $\forall n\in\omega$ $A$.<br />
* $\Delta^1_n(a)\subsetneq\Sigma^1_n(a)\subsetneq\Delta^1_{n+1}(a)$<br />
* $\Delta^1_n(a)\subsetneq\Pi^1_n(a)\subsetneq\Delta^1_{n+1}(a)$<br />
<br />
Sierpinski showed that every $\mathbf{\Sigma}^1_2$ set of reals is the union of $\aleph_1$ Borel (=$\mathbf{\Delta}^1_1$) sets. It follows that every $\mathbf{\Sigma}^1_2$ set of reals is either (at most) countable or has the cardinality of the continuum.<br />
<br />
''Shoenfield’s absoluteness theorem'' is the statement that every $\Sigma^1_2(a)$ or $\Pi^1_2(a)$ relation is absolute for every inner model of ZF+DC that contains $a$ (as an element). It follows that $\mathbf{\Sigma}^1_2$ and $\mathbf{\Pi}^1_2$ relations are absolute for $L$, and also that every $\Sigma^1_2(a)$ real (by taking $X=\omega$) is in $L[a]$, in particular every $\Sigma^1_2$ (or $\Pi^1_2$) real is constructible.<br />
The set of all constructible reals is $\Sigma^1_2$, and so is the canonical well-ordering $<_L$ of $L$. For $U$ a nonprincipal $\kappa$-complete [[ultrafilter]] on some [[measurable]] cardinal $\kappa$, then the collection of all sets of reals in $L[U]$ is $\Sigma^1_3$, and so is the canonical well-ordering $<_{L[U]}$ of $L[U]$. <br />
<br />
If [[zero sharp|$0^\#$]] exists then it is a $\Sigma^1_3$ real and the singleton $\{0^\#\}$ is a $\Pi^1_2$ set of reals. If for every real $a\in\omega^\omega$, the sharp $a^\#$ exists then every $\Sigma^1_3$ set of reals is the union of $\aleph_2$ Borel sets.<br />
<br />
== Regularity properties ==<br />
<br />
Let $A\subseteq(\omega^\omega)^k$ be a k-dimensional set of reals. We say that $A$ is ''null'' if it has [[:wikipedia:outer measure#Method I|outer measure]] 0. We say that $A$ is ''nowhere dense'' if its complement contains an open dense set, and that $A$ is ''meagre'' (or ''of first category'') if it is a countable union of nowhere dense set. Finally we say that $A$ is ''perfect'' if it has no isolated point.<br />
<br />
Then, we define the following ''regularity properties'':<br />
* $A$ is ''Lebesgue measurable'' if there exists a Borel set $B$ such that $A\Delta B$ is null.<br />
* $A$ has the ''Baire property'' if there exists an open set $B$ such that $A\Delta B$ is meagre.<br />
* $A$ has the ''perfect set property'' if it is either countable or has a perfect subset.<br />
Where $A\Delta B=(A\setminus B)\cup(B\setminus A)$ denotes symmetric difference. In ZFC there exists Lebesgue non-measurable sets without the Baire property nor the perfect set property, but it is interesting to see which projective sets have those regularity properties.<br />
<br />
In $L$ there is a $\Delta^1_2$ set of reals that is not Lebesgue measurable and has no perfect subset. Also there is a $\Pi^1_1$ set of reals without the perfect set property.<br />
<br />
If every $\Sigma^1_3$ set of reals is Lebesgue measurable then $\aleph_1$ is [[inaccessible]] in $L$.<br />
<br />
If $A$ is a $\Sigma^1_2(a)$ set of reals and contains a real that is not in $L[a]$ then $A$ has the perfect set property. Note that every uncountable set with the perfect set property has the cardinality of the continuum.<br />
<br />
The following statements are equivalent:<br />
* For every real $a$, $\aleph_1^{L[a]}$ is countable.<br />
* Every $\mathbf{\Pi}^1_1$ set has the perfect set property.<br />
* Every $\mathbf{\Sigma}^1_2$ set has the perfect set property.<br />
<br />
If $E$ is a $\mathbf{\Pi}^1_1$ equivalence relation on $\omega^\omega$ then either $E$ has at most $\aleph_0$ equivalence classes or there exists a perfect set of mutually inequivalent reals. If $E$ is a $\mathbf{\Sigma}^1_1$ equivalence relation on $\omega^\omega$ then either $E$ has at most $\aleph_1$ equivalence classes or there exists a perfect set of mutually inequivalent reals.<br />
<br />
=== Prewellordering, scale and uniformization properties ===<br />
<br />
A ''norm'' on a set $A$ is a function $\varphi:A\to Ord$ from $A$ to the ordinals. A ''prewellordering'' is a relation $\preceq$ that is like a well-ordering except we do not require it to be reflexive or antisymmetric. If $\preceq$ is a prewellordering then the $a\equiv b\iff(a\preceq b\land b\preceq a)$ is an equivalence relation, and $\preceq$ is a well-ordering of the equivalence classes of $\equiv$. If $\varphi$ is a norm then $a\preceq_\varphi b\iff\varphi(a)\leq\varphi(b)$ is a prewellordering.<br />
<br />
A pointclass $\Gamma$ has the ''prewellordering property'' if every set $A$ in $\Gamma$ has a $\Gamma$-norm: a norm $\varphi:A\to Ord$, a $\Gamma$ relation $P(x,y)$ and a $\neg\Gamma$ relation $Q(x,y)$ such that for every $y\in A$ and all $x$: $x\in A\land\varphi(x)\leq\varphi(y)\iff P(x,y)\iff Q(x,y)$. The pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the prewellordering property.<br />
<br />
A ''scale'' on $A$ is a sequence of norms $\{\varphi_n:n\in\omega\}$ such that for every sequence of points $\{x_i:i\in\omega\}$, if for every $n$ the sequence $\{\varphi_n(x_i):n\in\omega\}$ is eventually constant with value $\alpha_n$, then $(lim_{i\to\omega}$ $x_i)\in A$ and for every $n$, $\varphi_n(lim_{i\to\omega}$ $x_i)\leq\alpha_n$. A pointclass $\Gamma$ has the ''scale property'' if for every set $A$ in $\Gamma$, there exists a $\Gamma$-scale for $A$: a scale $\{\varphi_n:n\in\omega\}$, a $\Gamma$ relation $P(n,x,y)$ and a $\neg\Gamma$ relation $Q(x,y)$ such that for every $n$, every $y\in A$ and all $x$: $x\in A\land\varphi_n(x)\leq\varphi_n(y)\iff P(n,x,y)\iff Q(n,x,y)$. Again, the pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the scale property.<br />
<br />
A set $A\subseteq\omega^\omega\times\omega^\omega$ is ''uniformized'' by a function $F$ if $dom(F)=\{x\in\omega^\omega:\exists y\in\omega^\omega$ $(x,y)\in A\}$ and $(x,F(x))\in A$ for all $x\in dom(F)$. A pointclass $\Gamma$ has the ''uniformization'' property if every set in $\Gamma$ can be uniformized by a function in $\Gamma$. The pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the uniformization property.<br />
<br />
=== Reduction and separation properties ===<br />
<br />
For any four sets $A$, $B$, $A'$ and $B'$, $(A',B')$ ''reduces'' $(A,B)$ if $A'\subseteq A$, $B'\subseteq B$, $A'\cup B'=A\cup B$, but $A'\cap B'=\empty$. Thus $A'$ and $B'$ partition $A\cup B$. A pointclass $\Gamma$ has the ''reduction property'' if for every $A$, $B$ in $\Gamma$ there exists $A'$, $B'$ in $\Gamma$ such that $(A',B')$ reduces $(A,B)$. $\Pi^1_1(a)$ has the reduction property for every $a\in\omega^\omega$.<br />
<br />
A pointclass $\Gamma$ has the ''separation property'' if for any disjoint subsets $A$, $B$ of $(\omega^\omega)^k$ for some $k$, if $A$ and $B$ are in $\Gamma$ then there is a $C$ in $\Delta_\Gamma$ such that $A\subseteq C$ and $B\cap C=\empty$. $\Sigma^1_1(a)$ has the separation property for every $a\in\omega^\omega$.<br />
<br />
If $\Gamma$ has the reduction property, then $\neg\Gamma$ has the separation property but not the reduction property. It is impossible for $\Gamma$ to have both the reduction and the separation properties. Every pointclass with the prewellordering property has the reduction property. Thus it is impossible for both a pointclass and its complement to have the prewellordering or scale properties.<br />
<br />
== Projective determinacy ==<br />
<br />
''See also: [[axiom of determinacy]]''<br />
<br />
''Determinacy'' is a kind of regularity property. For every set of reals $A$, the game $G_A$ is the infinite game of perfect information of length $\omega$ where both players constructs a sequence (i.e. a real) by playing elements of $\omega$, one after the other, such that the first player's goal is to have the constructed real be in $A$, and the second player's goal is to have the constructed real be in $A$'s complement. $A$ is ''determined'' if the game $G_A$ is determined, i.e. one of the two players have a winning strategy for $G_A$.<br />
<br />
Given a pointclass $\Gamma$, ''$\Gamma$-determinacy'' is the statement "every $A\in\Gamma$ is determined". $\Gamma$-determinacy and $\neg\Gamma$-determinacy are always equivalent. $\omega^\omega$-determinacy is the ''axiom of determinacy'' and is implied false by the [[axiom of choice]]. The '''axiom of projective determinacy''' (PD) is precisely $(\bigcup_{n\in\omega}\mathbf{\Sigma}^1_n)$-determinacy. Given some class $M$ (e.g. $OD$, $L(\mathbb{R})$, ...), ''$M$-determinacy'' is an abbreviation for $(M\cap\mathcal{P}(\omega^\omega))$-determinacy. $L(\mathbb{R})$-determinacy notably follows from large cardinal axioms, in particular the existence of infinitely many [[Woodin]] cardinals with a [[measurable]] above them all.<br />
<br />
Martin showed that ZFC alone is sufficient to prove Borel determinacy (i.e. $\mathbf{\Delta}^1_1$-determinacy). However, for every $a\in\omega^\omega$, $\Sigma^1_1(a)$-determinacy is equivalent to "the sharp $a^\#$ exists", thus Borel determinacy is the best result possible in ZFC alone. Analytic (i.e. $\mathbf{\Sigma^1_1}$) determinacy follows from the existence of a measurable cardinal, or even just of a [[Ramsey]] cardinal. Stronger forms of projective determinacies requires considerably stronger large cardinal axioms: for every $n$, $\mathbf{\Delta}^1_{n+1}$-determinacy implies the existence of an inner model with $n$ Woodin cardinals.<br />
<br />
Note that for every $n$, $\mathbf{\Sigma}^1_n$-determinacy is equivalent to $\mathbf{\Pi}^1_n$-determinacy. Furthertmore, under DC (the ''[[:wikipedia:axiom of dependent choice|axiom of dependent choice]]'') for every $n\in\omega$, $\mathbf{\Delta}^1_{2n}$-determinacy is equivalent to $\mathbf{\Sigma}^1_{2n}$-determinacy ($\mathbf{\Pi}^1_{2n}$-determinacy)<br />
<br />
Assume $\mathbf{\Sigma}^1_n$ (or $\mathbf{\Pi}^1_n$) determinacy and that the axiom of choice holds for ''countable'' sets of reals (which follows from DC). Then every $\mathbf{\Sigma}^1_{n+1}$ set of reals is Lebesgue measurable, has the Baire property and has the perfect set property.<br />
<br />
Assume projective determinacy; then the following pointclasses have the reduction, prewellordering, scale and uniformization properties, for every $a\in\omega^\omega$: $\Pi^1_1(a), \Sigma^1_2(a), \Pi^1_3(a), ..., \Pi^1_{2n+1}(a), \Sigma^1_{2n+2}(a), ...$ This is known as the ''periodicity theorem''. On the other hand, if $L[U]$ contains every real for some nonprincipal $\kappa$-complete [[ultrafilter]] $U$ on a measurable cardinal $\kappa$, then every $\Sigma^1_n(a)$ has the reduction and prewellordering properties for $n\geq 2$ and every $a\in\omega^\omega$.<br />
<br />
=== Projective ordinals ===<br />
<br />
For every pointclass $\Gamma$, define $\delta_\Gamma$ as the supremum of the length of $\Gamma$ prewellorderings of $\omega^\omega$. We then define the ''projective ordinals'' to be $\delta^1_n=\delta_{\mathbf{\Sigma}^1_n}=\delta_{\mathbf{\Pi}^1_n}$. It can be shown without AD that $\delta^1_1=\omega_1$ and that $\delta^1_2\leq\omega_2$. Under AD, each projective ordinal is a regular cardinal and the sequence $\{\delta^1_n:n\in\omega\}$ is a strictly increasing sequence of measurable cardinals, also $\delta^1_2=\omega_2$, $\delta^1_3=\omega_{\omega+1}$ and $\delta^1_4=\omega_{\omega+2}$. In general, $\delta^1_{2n+2}\leq(\delta^1_{2n+1})^{+}$. Under DC this becomes an equality, also every $\delta^1_{2n+1}$ is the successor of a cardinal of cofinality $\omega$.<br />
<br />
Define $E:\omega\to\omega_1$ by recursion the following way: $E(0)=1$, $E(n+1)=\omega^{E(n)}$ (ordinal exponentiation). Then, under AD+DC, one have $\delta^1_{2n+3}=\omega_{E(2n+1)+1}$, also every $\delta^1_{2n+3}$ has the strong partition property $\delta^1_{2n+3}\to(\delta^1_{2n+3})^{\delta^1_{2n+3}}_\alpha$ for every $\alpha<\delta^1_{2n+3}$.<br />
<br />
Let's say a set of reals $A$ is $\gamma$-Borel (for a cardinal $\gamma$) if it is in the smallest collection of sets containing all closed sets of $(\omega^\omega)^k$ that is closed under complementations and unions of less than $\gamma$ sets. If $\gamma$ is not a cardinal then $A$ is $\gamma$-Borel if it is $\gamma^{+}$-Borel where $\gamma^{+}$ is the smallest cardinal >$\gamma$. Note that a set is Borel if and only if it is $\aleph_1$-Borel.<br />
<br />
Assume $\mathbf{\Delta}^1_{2n}$-determinacy; then a set of reals $A$ is $\mathbf{\Delta}^1_{2n+1}$ if and only if it is $\delta^1_{2n+1}$-Borel. Now, assume AD+DC; then a set $A$ is $\mathbf{\Sigma}^1_{2n+2}$ if and only if it is the union of $\delta^1_{2n+1}$-many sets that are $\mathbf{\Delta}^1_{2n+1}$.<br />
<br />
=== Projective determinacy from large cardinals ===<br />
<br />
Woodin showed that $\mathbf{\Pi}^1_{n+1}$-determinacy follows from the existence of $n$ [[Woodin]] cardinals with a measurable above them all, and projective determinacy thus follows from the existence of infinitely many Woodin cardinals. He also showed that $\mathbf{\Pi}^1_2$-determinacy is equivalent to "for all $x\in\mathbb{R}$, there is a countable ordinal $\delta$ such that $\delta$ is a Woodin cardinal in some inner model of ZFC containing $x$, and that $\mathbf{\Delta}^1_2$-determinacy is equivalent to "for every $x\in\mathbb{R}$, there is an inner model M such that $x\in M$ and $M\models ZFC+$"there is a Woodin cardinal". <cite>KoellnerWoodin2010:LCFD</cite><br />
<br />
ZFC + (lightface) $\Delta^1_2$-determinacy implies that there many $x$ such that HOD$^{L[x]}$ is a model of ZFC+"$\omega_2^{L[x]}$ is a Woodin cardinal".<br />
Z$_2$+$\Delta^1_2$-determinacy is conjectured to be equiconsistent with ZFC+"Ord is Woodin", where "Ord is Woodin" is expressed as an axiom scheme and Z$_2$ is [[:wikipedia:second-order arithmetic|second-order arithmetic]].<br />
Z$_3$+$\Delta^1_2$-determinacy is provably equiconsistent with NBG+"Ord is Woodin" where NBG is [[:wikipedia:Von Neumann–Bernays–Gödel set theory|Von Neumann–Bernays–Gödel set theory]] and $Z_3$ is third-order arithmetic.<br />
<br />
Gitik and Schindler showed that, in ZF, if $\aleph_\omega$ is a strong limit cardinal and $2^{\aleph_\omega}>\aleph_{\omega_1}$, then the axiom of projective determinacy holds. Also, if there is a singular cardinal of uncountable cofinality such that the sets of the cardinals below it such that the GCH holds is both [[stationary]] and costationary, then again the axiom of projective determinacy holds. It is not known whether these two results extends to $L(\mathbb{R})$-determinacy. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
Foreman, Magidor and Schindler showed that if there exists infinitely many cardinals $\delta$ above the continuum such that both $\delta$ and $\delta^{+}$ have the [[tree property]], then the axiom of projective determinacy holds. This hypothesis was shown to be consistent relative to the existence of infinitely many [[supercompact]] cardinals by James Cummings and Foreman. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Projective&diff=2063Projective2017-11-11T20:01:54Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: Projective sets and determinacy}}<br />
We say that $\Gamma$ is a ''pointclass'' if it is a collection of subsets of a [[:wikipedia:Polish space|Polish space]]. <br />
The '''lightface and bolface projective hierarchies''' are hierarchies of pointclasses of some Polish space $X$ defined by repeated applications of projections and complementations from either recursively enumerable or closed sets respectively.<br />
<br />
''Most results in this article can be found in <cite>Jech2003:SetTheory</cite> and <cite>Kanamori2009:HigherInfinite</cite> unless indicated otherwise.''<br />
<br />
== Definitions ==<br />
<br />
The following definitions are made by taking $X=\omega^\omega$, the ''Baire space'', i.e. the set of all functions $f:\mathbb{N}\to\mathbb{N}$. We will identify its members with the corresponding real numbers under some fixed bijection between $\mathbb{R}$ and $\omega^\omega$. The definitions presented here can be easily extended to other Polish spaces than the Baire space.<br />
<br />
Let $\mathbf{\Sigma}^0_1$ be the pointclass that contains all open subsets of the Polish space $\omega^\omega$. Let $\mathbf{\Pi}^0_1$ be the pointclass containing the complements of the $\mathbf{\Sigma}^0_1$ sets.<br />
<br />
We define the '''boldface projective pointclasses''' $\mathbf{\Sigma}^1_n$, $\mathbf{\Pi}^1_n$ and $\mathbf{\Delta}^1_n$ the following way:<br />
# $\mathbf{\Sigma}^1_1$ contains all the images of $\mathbf{\Pi}^0_1$ sets by continuous functions; its members are called the ''analytic sets''.<br />
# Now, for all $n$, define $\mathbf{\Pi}^1_n$ to be the set of the complements of the $\mathbf{\Sigma}^1_n$ sets; the members of $\mathbf{\Pi}^1_1$ are called the ''coanalytic sets''.<br />
# For all $n$, define $\mathbf{\Sigma}^1_{n+1}$ to be the set of the images of $\mathbf{\Pi}^1_n$ sets by continuous functions.<br />
# Finally, let $\mathbf{\Delta}^1_n=\mathbf{\Sigma}^1_n\cap\mathbf{\Pi}^1_n$. The members of $\mathbf{\Delta}^1_1$ are the ''Borel'' sets.<br />
<br />
The '''relativized lightface projective pointclasses''' $\Sigma^1_n(a)$, $\Pi^1_n(a)$ and $\Delta^1_n(a)$ (for $a\in\omega^\omega$) are defined similarly except that $\Sigma^1_1(a)$ is defined as the set of all $A\subseteq\omega^\omega$ such that $A=\{x\in\omega^\omega:\exists y\in\omega^\omega$ $\exists n\in\omega$ $R(x\restriction n,y\restriction n,a\restriction n)\}$, that is, $A$ is recursively definable by a formula with only existential quantifiers ranging on members of $\omega^\omega$ or on $\omega$ and whose only parameter is $a$.<br />
<br />
The (non-relativized) '''lightface projective classes''', also known as ''analytical pointclasses'', are the special cases $\Sigma^1_n$, $\Pi^1_n$ and $\Delta^1_n$ of relativized lightface projective pointclasses where $a=\empty$. Let $\Sigma^0_1$ be the pointclass of all ''recursively enumerable'' sets, i.e. the sets $A$ such there exists a recursive relation $R$ such that $A=\{x\in\omega^\omega:\exists n\in\omega$ $R(x\restriction n)\}$, and $\Pi^0_1$ contain the completements of $\Sigma^0_1$ sets. Then the $\Sigma^1_1$ sets are precisely the projections of $\Pi^0_1$ sets.<br />
<br />
Given an arbitrary pointclass $\Gamma$, define $\neg\Gamma$ as the set of the complements of $\Gamma$'s elements, for example $\Pi^1_n(a)=\neg\Sigma^1_n(a)$. Also let $\Delta_\Gamma=\Gamma\cap\neg\Gamma$, for example $\Delta^1_n(a)=\Delta_{\Pi^1_n(a)}=\Delta_{\Sigma^1_n(a)}$.<br />
<br />
== Properties ==<br />
<br />
Every $\mathbf{\Sigma}^1_n$ set is $\Sigma^1_n(a)$ for some $a\in\omega^\omega$, in fact $\mathbf{\Sigma}^1_n=\bigcup_{a\in\omega^\omega}\Sigma^1_n(a)$. A similar statement holds for $\mathbf{\Pi}^1_n$ sets and $\mathbf{\Delta}^1_n$ sets. This means the boldface projective sets are precisely the set definable using only real and arithmetical quantifiers and real parameters.<br />
<br />
The following statements also holds when replacing relativized lightface pointclasses by their boldface counterparts:<br />
* If $A$ and $B$ are $\Sigma^1_n(a)$ relations, then so are $\exists x\in\omega^\omega$ $A$, $A\land B$, $A\lor B$, $\exists n\in\omega$ $A$ and $\forall n\in\omega$ $A$.<br />
* If $A$ and $B$ are $\Pi^1_n(a)$ relations, then so are $\forall x\in\omega^\omega$ $A$, $A\land B$, $A\lor B$, $\exists n\in\omega$ $A$ and $\forall n\in\omega$ $A$.<br />
* If $A$ is a $\Sigma^1_n(a)$ relation then $\neg A$ is a $\Pi^1_n(a)$ relation. If $A$ is $\Pi^1_n(a)$ then $\neg A$ is $\Sigma^1_n(a)$.<br />
* If $A$ is a $\Sigma^1_n(a)$ relation and $B$ is a $\Pi^1_n(a)$ relation, then $A\Rightarrow B$ is a $\Pi^1_n(a)$ relation.<br />
* If $A$ is a $\Pi^1_n(a)$ relation and $B$ is a $\Sigma^1_n(a)$ relation, then $A\Rightarrow B$ is a $\Sigma^1_n(a)$ relation.<br />
* If $A$ and $B$ are $\Delta^1_n(a)$, then so are $\neg A$, $A\land B$, $A\lor B$, $A\Rightarrow B$, $A\Leftrightarrow B$, $\exists n\in\omega$ $A$, $\forall n\in\omega$ $A$.<br />
* $\Delta^1_n(a)\subsetneq\Sigma^1_n(a)\subsetneq\Delta^1_{n+1}(a)$<br />
* $\Delta^1_n(a)\subsetneq\Pi^1_n(a)\subsetneq\Delta^1_{n+1}(a)$<br />
<br />
Sierpinski showed that every $\mathbf{\Sigma}^1_2$ set of reals is the union of $\aleph_1$ Borel (=$\mathbf{\Delta}^1_1$) sets. It follows that every $\mathbf{\Sigma}^1_2$ set of reals is either (at most) countable or has the cardinality of the continuum.<br />
<br />
''Shoenfield’s absoluteness theorem'' is the statement that every $\Sigma^1_2(a)$ or $\Pi^1_2(a)$ relation is absolute for every inner model of ZF+DC that contains $a$ (as an element). It follows that $\mathbf{\Sigma}^1_2$ and $\mathbf{\Pi}^1_2$ relations are absolute for $L$, and also that every $\Sigma^1_2(a)$ real (by taking $X=\omega$) is in $L[a]$, in particular every $\Sigma^1_2$ (or $\Pi^1_2$) real is constructible.<br />
The set of all constructible reals is $\Sigma^1_2$, and so is the canonical well-ordering $<_L$ of $L$. For $U$ a nonprincipal $\kappa$-complete [[ultrafilter]] on some [[measurable]] cardinal $\kappa$, then the collection of all sets of reals in $L[U]$ is $\Sigma^1_3$, and so is the canonical well-ordering $<_{L[U]}$ of $L[U]$. <br />
<br />
If [[zero sharp|$0^\#$]] exists then it is a $\Sigma^1_3$ real and the singleton $\{0^\#\}$ is a $\Pi^1_2$ set of reals. If for every real $a\in\omega^\omega$, the sharp $a^\#$ exists then every $\Sigma^1_3$ set of reals is the union of $\aleph_2$ Borel sets.<br />
<br />
== Regularity properties ==<br />
<br />
Let $A\subseteq(\omega^\omega)^k$ be a k-dimensional set of reals. We say that $A$ is ''null'' if it has [[:wikipedia:outer measure#Method I|outer measure]] 0. We say that $A$ is ''nowhere dense'' if its complement contains an open dense set, and that $A$ is ''meagre'' (or ''of first category'') if it is a countable union of nowhere dense set. Finally we say that $A$ is ''perfect'' if it has no isolated point.<br />
<br />
Then, we define the following ''regularity properties'':<br />
* $A$ is ''Lebesgue measurable'' if there exists a Borel set $B$ such that $A\Delta B$ is null.<br />
* $A$ has the ''Baire property'' if there exists an open set $B$ such that $A\Delta B$ is meagre.<br />
* $A$ has the ''perfect set property'' if it is either countable or has a perfect subset.<br />
Where $A\Delta B=(A\setminus B)\cup(B\setminus A)$ denotes symmetric difference. In ZFC there exists Lebesgue non-measurable sets without the Baire property nor the perfect set property, but it is interesting to see which projective sets have those regularity properties.<br />
<br />
In $L$ there is a $\Delta^1_2$ set of reals that is not Lebesgue measurable and has no perfect subset. Also there is a $\Pi^1_1$ set of reals without the perfect set property.<br />
<br />
If every $\Sigma^1_3$ set of reals is Lebesgue measurable then $\aleph_1$ is [[inaccessible]] in $L$.<br />
<br />
If $A$ is a $\Sigma^1_2(a)$ set of reals and contains a real that is not in $L[a]$ then $A$ has the perfect set property. Note that every uncountable set with the perfect set property has the cardinality of the continuum.<br />
<br />
The following statements are equivalent:<br />
* For every real $a$, $\aleph_1^{L[a]}$ is countable.<br />
* Every $\mathbf{\Pi}^1_1$ set has the perfect set property.<br />
* Every $\mathbf{\Sigma}^1_2$ set has the perfect set property.<br />
<br />
If $E$ is a $\mathbf{\Pi}^1_1$ equivalence relation on $\omega^\omega$ then either $E$ has at most $\aleph_0$ equivalence classes or there exists a perfect set of mutually inequivalent reals. If $E$ is a $\mathbf{\Sigma}^1_1$ equivalence relation on $\omega^\omega$ then either $E$ has at most $\aleph_1$ equivalence classes or there exists a perfect set of mutually inequivalent reals.<br />
<br />
=== Prewellordering, scale and uniformization properties ===<br />
<br />
A ''norm'' on a set $A$ is a function $\varphi:A\to Ord$ from $A$ to the ordinals. A ''prewellordering'' is a relation $\preceq$ that is like a well-ordering except we do not require it to be reflexive or antisymmetric. If $\preceq$ is a prewellordering then the $a\equiv b\iff(a\preceq b\land b\preceq a)$ is an equivalence relation, and $\preceq$ is a well-ordering of the equivalence classes of $\equiv$. If $\varphi$ is a norm then $a\preceq_\varphi b\iff\varphi(a)\leq\varphi(b)$ is a prewellordering.<br />
<br />
A pointclass $\Gamma$ has the ''prewellordering property'' if every set $A$ in $\Gamma$ has a $\Gamma$-norm: a norm $\varphi:A\to Ord$, a $\Gamma$ relation $P(x,y)$ and a $\neg\Gamma$ relation $Q(x,y)$ such that for every $y\in A$ and all $x$: $x\in A\land\varphi(x)\leq\varphi(y)\iff P(x,y)\iff Q(x,y)$. The pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the prewellordering property.<br />
<br />
A ''scale'' on $A$ is a sequence of norms $\{\varphi_n:n\in\omega\}$ such that for every sequence of points $\{x_i:i\in\omega\}$, if for every $n$ the sequence $\{\varphi_n(x_i):n\in\omega\}$ is eventually constant with value $\alpha_n$, then $(lim_{i\to\omega}$ $x_i)\in A$ and for every $n$, $\varphi_n(lim_{i\to\omega}$ $x_i)\leq\alpha_n$. A pointclass $\Gamma$ has the ''scale property'' if for every set $A$ in $\Gamma$, there exists a $\Gamma$-scale for $A$: a scale $\{\varphi_n:n\in\omega\}$, a $\Gamma$ relation $P(n,x,y)$ and a $\neg\Gamma$ relation $Q(x,y)$ such that for every $n$, every $y\in A$ and all $x$: $x\in A\land\varphi_n(x)\leq\varphi_n(y)\iff P(n,x,y)\iff Q(n,x,y)$. Again, the pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the scale property.<br />
<br />
A set $A\subseteq\omega^\omega\times\omega^\omega$ is ''uniformized'' by a function $F$ if $dom(F)=\{x\in\omega^\omega:\exists y\in\omega^\omega$ $(x,y)\in A\}$ and $(x,F(x))\in A$ for all $x\in dom(F)$. A pointclass $\Gamma$ has the ''uniformization'' property if every set in $\Gamma$ can be uniformized by a function in $\Gamma$. The pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the uniformization property.<br />
<br />
=== Reduction and separation properties ===<br />
<br />
For any four sets $A$, $B$, $A'$ and $B'$, $(A',B')$ ''reduces'' $(A,B)$ if $A'\subseteq A$, $B'\subseteq B$, $A'\cup B'=A\cup B$, but $A'\cap B'=\empty$. Thus $A'$ and $B'$ partition $A\cup B$. A pointclass $\Gamma$ has the ''reduction property'' if for every $A$, $B$ in $\Gamma$ there exists $A'$, $B'$ in $\Gamma$ such that $(A',B')$ reduces $(A,B)$. $\Pi^1_1(a)$ has the reduction property for every $a\in\omega^\omega$.<br />
<br />
A pointclass $\Gamma$ has the ''separation property'' if for any disjoint subsets $A$, $B$ of $(\omega^\omega)^k$ for some $k$, if $A$ and $B$ are in $\Gamma$ then there is a $C$ in $\Delta_\Gamma$ such that $A\subseteq C$ and $B\cap C=\empty$. $\Sigma^1_1(a)$ has the separation property for every $a\in\omega^\omega$.<br />
<br />
If $\Gamma$ has the reduction property, then $\neg\Gamma$ has the separation property but not the reduction property. It is impossible for $\Gamma$ to have both the reduction and the separation properties. Every pointclass with the prewellordering property has the reduction property. Thus it is impossible for both a pointclass and its complement to have the prewellordering or scale properties.<br />
<br />
== Projective determinacy ==<br />
<br />
''See also: [[axiom of determinacy]]''<br />
<br />
''Determinacy'' is a kind of regularity property. For every set of reals $A$, the game $G_A$ is the infinite game of perfect information of length $\omega$ where both players constructs a sequence (i.e. a real) by playing elements of $\omega$, one after the other, such that the first player's goal is to have the constructed real be in $A$, and the second player's goal is to have the constructed real be in $A$'s complement. $A$ is ''determined'' if the game $G_A$ is determined, i.e. one of the two players have a winning strategy for $G_A$.<br />
<br />
Given a pointclass $\Gamma$, ''$\Gamma$-determinacy'' is the statement "every $A\in\Gamma$ is determined". $\Gamma$-determinacy and $\neg\Gamma$-determinacy are always equivalent. $\omega^\omega$-determinacy is the ''axiom of determinacy'' and is implied false by the [[axiom of choice]]. The '''axiom of projective determinacy''' ($PD$) is precisely $(\bigcup_{n\in\omega}\mathbf{\Sigma}^1_n)$-determinacy. Given some class $M$ (e.g. $OD$, $L(\mathbb{R})$, ...), ''$M$-determinacy'' is an abbreviation for $(M\cap\mathcal{P}(\omega^\omega))$-determinacy. $L(\mathbb{R})$-determinacy notably follows from large cardinal axioms, in particular the existence of infinitely many [[Woodin]] cardinals with a [[measurable]] above them all.<br />
<br />
Martin showed that ZFC alone is sufficient to prove Borel determinacy (i.e. $\mathbf{\Delta}^1_1$-determinacy). However, for every $a\in\omega^\omega$, $\Sigma^1_1(a)$-determinacy is equivalent to "the sharp $a^\#$ exists", thus Borel determinacy is the best result possible in ZFC alone. Analytic (i.e. $\mathbf{\Sigma^1_1}$) determinacy follows from the existence of a measurable cardinal, or even just of a [[Ramsey]] cardinal. Stronger forms of projective determinacies requires considerably stronger large cardinal axioms: for every $n$, $\mathbf{\Delta}^1_{n+1}$-determinacy implies the existence of an inner model with $n$ Woodin cardinals.<br />
<br />
Note that for every $n$, $\mathbf{\Sigma}^1_n$-determinacy is equivalent to $\mathbf{\Pi}^1_n$-determinacy. Furthertmore, under $DC$ (the ''[[:wikipedia:axiom of dependent choice|axiom of dependent choice]]'') for every $n\in\omega$, $\mathbf{\Delta}^1_{2n}$-determinacy is equivalent to $\mathbf{\Sigma}^1_{2n}$-determinacy ($\mathbf{\Pi}^1_{2n}$-determinacy)<br />
<br />
Assume $\mathbf{\Sigma}^1_n$ (or $\mathbf{\Pi}^1_n$) determinacy and that the axiom of choice holds for ''countable'' sets of reals (which follows from $DC$). Then every $\mathbf{\Sigma}^1_{n+1}$ set of reals is Lebesgue measurable, has the Baire property and has the perfect set property.<br />
<br />
Assume projective determinacy; then the following pointclasses have the reduction, prewellordering, scale and uniformization properties, for every $a\in\omega^\omega$: $\Pi^1_1(a), \Sigma^1_2(a), \Pi^1_3(a), ..., \Pi^1_{2n+1}(a), \Sigma^1_{2n+2}(a), ...$ This is known as the ''periodicity theorem''. On the other hand, if $L[U]$ contains every real for some nonprincipal $\kappa$-complete [[ultrafilter]] $U$ on a measurable cardinal $\kappa$, then every $\Sigma^1_n(a)$ has the reduction and prewellordering properties for $n\geq 2$ and every $a\in\omega^\omega$.<br />
<br />
=== Projective ordinals ===<br />
<br />
For every pointclass $\Gamma$, define $\delta_\Gamma$ as the supremum of the length of $\Gamma$ prewellorderings of $\omega^\omega$. We then define the ''projective ordinals'' to be $\delta^1_n=\delta_{\mathbf{\Sigma}^1_n}=\delta_{\mathbf{\Pi}^1_n}$. It can be shown without AD that $\delta^1_1=\omega_1$ and that $\delta^1_2\leq\omega_2$. Under AD, each projective ordinal is a regular cardinal and the sequence $\{\delta^1_n:n\in\omega\}$ is a strictly increasing sequence of measurable cardinals, also $\delta^1_2=\omega_2$, $\delta^1_3=\omega_{\omega+1}$ and $\delta^1_4=\omega_{\omega+2}$. In general, $\delta^1_{2n+2}\leq(\delta^1_{2n+1})^{+}$. Under $DC$ this becomes an equality, also every $\delta^1_{2n+1}$ is the successor of a cardinal of cofinality $\omega$.<br />
<br />
Define $E:\omega\to\omega_1$ by recursion the following way: $E(0)=1$, $E(n+1)=\omega^{E(n)}$ (ordinal exponentiation). Then, under AD+DC, one have $\delta^1_{2n+3}=\omega_{E(2n+1)+1}$, also every $\delta^1_{2n+3}$ has the strong partition property $\delta^1_{2n+3}\to(\delta^1_{2n+3})^{\delta^1_{2n+3}}_\alpha$ for every $\alpha<\delta^1_{2n+3}$.<br />
<br />
Let's say a set of reals $A$ is $\gamma$-Borel (for a cardinal $\gamma$) if it is in the smallest collection of sets containing all closed sets of $(\omega^\omega)^k$ that is closed under complementations and unions of less than $\gamma$ sets. If $\gamma$ is not a cardinal then $A$ is $\gamma$-Borel if it is $\gamma^{+}$-Borel where $\gamma^{+}$ is the smallest cardinal >$\gamma$. Note that a set is Borel if and only if it is $\aleph_1$-Borel.<br />
<br />
Assume $\mathbf{\Delta}^1_{2n}$-determinacy; then a set of reals $A$ is $\mathbf{\Delta}^1_{2n+1}$ if and only if it is $\delta^1_{2n+1}$-Borel. Now, assume AD+DC; then a set $A$ is $\mathbf{\Sigma}^1_{2n+2}$ if and only if it is the union of $\delta^1_{2n+1}$-many sets that are $\mathbf{\Delta}^1_{2n+1}$.<br />
<br />
=== Projective determinacy from large cardinals ===<br />
<br />
Woodin showed that $\mathbf{\Pi}^1_{n+1}$-determinacy follows from the existence of $n$ [[Woodin]] cardinals with a measurable above them all, and projective determinacy thus follows from the existence of infinitely many Woodin cardinals. He also showed that $\mathbf{\Pi}^1_2$-determinacy is equivalent to "for all $x\in\mathbb{R}$, there is a countable ordinal $\delta$ such that $\delta$ is a Woodin cardinal in some inner model of ZFC containing $x$, and that $\mathbf{\Delta}^1_2$-determinacy is equivalent to "for every $x\in\mathbb{R}$, there is an inner model M such that $x\in M$ and $M\models ZFC+$"there is a Woodin cardinal". <cite>KoellnerWoodin2010:LCFD</cite><br />
<br />
ZFC + (lightface) $\Delta^1_2$-determinacy implies that there many $x$ such that HOD$^{L[x]}$ is a model of ZFC+"$\omega_2^{L[x]}$ is a Woodin cardinal".<br />
Z$_2$+$\Delta^1_2$-determinacy is conjectured to be equiconsistent with ZFC+"Ord is Woodin", where "Ord is Woodin" is expressed as an axiom scheme and Z$_2$ is [[:wikipedia:second-order arithmetic|second-order arithmetic]].<br />
Z$_3$+$\Delta^1_2$-determinacy is provably equiconsistent with NBG+"Ord is Woodin" where NBG is [[:wikipedia:Von Neumann–Bernays–Gödel set theory|Von Neumann–Bernays–Gödel set theory]] and $Z_3$ is third-order arithmetic.<br />
<br />
Gitik and Schindler showed that, in ZF, if $\aleph_\omega$ is a strong limit cardinal and $2^{\aleph_\omega}>\aleph_{\omega_1}$, then the axiom of projective determinacy holds. Also, if there is a singular cardinal of uncountable cofinality such that the sets of the cardinals below it such that the GCH holds is both [[stationary]] and costationary, then again the axiom of projective determinacy holds. It is not known whether these two results extends to $L(\mathbb{R})$-determinacy. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
Foreman, Magidor and Schindler showed that if there exists infinitely many cardinals $\delta$ above the continuum such that both $\delta$ and $\delta^{+}$ have the [[tree property]], then the axiom of projective determinacy holds. This hypothesis was shown to be consistent relative to the existence of infinitely many [[supercompact]] cardinals by James Cummings and Foreman. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Projective&diff=2062Projective2017-11-11T19:59:23Z<p>Wabb2t: /* Projective determinacy from large cardinals */</p>
<hr />
<div>{{DISPLAYTITLE: Projective sets and determinacy}}<br />
We say that $\Gamma$ is a ''pointclass'' if it is a collection of subsets of a [[:wikipedia:Polish space|Polish space]]. <br />
The '''lightface and bolface projective hierarchies''' are hierarchies of pointclasses of some Polish space $X$ defined by repeated applications of projections and complementations from either recursively enumerable or closed sets respectively.<br />
<br />
''Most results in this article can be found in <cite>Jech2003:SetTheory</cite> and <cite>Kanamori2009:HigherInfinite</cite> unless indicated otherwise.''<br />
<br />
== Definitions ==<br />
<br />
The following definitions are made by taking $X=\omega^\omega$, the ''Baire space'', i.e. the set of all functions $f:\mathbb{N}\to\mathbb{N}$. We will identify its members with the corresponding real numbers under some fixed bijection between $\mathbb{R}$ and $\omega^\omega$. The definitions presented here can be easily extended to other Polish spaces than the Baire space.<br />
<br />
Let $\mathbf{\Sigma}^0_1$ be the pointclass that contains all open subsets of the Polish space $\omega^\omega$. Let $\mathbf{\Pi}^0_1$ be the pointclass containing the complements of the $\mathbf{\Sigma}^0_1$ sets.<br />
<br />
We define the '''boldface projective pointclasses''' $\mathbf{\Sigma}^1_n$, $\mathbf{\Pi}^1_n$ and $\mathbf{\Delta}^1_n$ the following way:<br />
# $\mathbf{\Sigma}^1_1$ contains all the images of $\mathbf{\Pi}^0_1$ sets by continuous functions; its members are called the ''analytic sets''.<br />
# Now, for all $n$, define $\mathbf{\Pi}^1_n$ to be the set of the complements of the $\mathbf{\Sigma}^1_n$ sets; the members of $\mathbf{\Pi}^1_1$ are called the ''coanalytic sets''.<br />
# For all $n$, define $\mathbf{\Sigma}^1_{n+1}$ to be the set of the images of $\mathbf{\Pi}^1_n$ sets by continuous functions.<br />
# Finally, let $\mathbf{\Delta}^1_n=\mathbf{\Sigma}^1_n\cap\mathbf{\Pi}^1_n$. The members of $\mathbf{\Delta}^1_1$ are the ''Borel'' sets.<br />
<br />
The '''relativized lightface projective pointclasses''' $\Sigma^1_n(a)$, $\Pi^1_n(a)$ and $\Delta^1_n(a)$ (for $a\in\omega^\omega$) are defined similarly except that $\Sigma^1_1(a)$ is defined as the set of all $A\subseteq\omega^\omega$ such that $A=\{x\in\omega^\omega:\exists y\in\omega^\omega$ $\exists n\in\omega$ $R(x\restriction n,y\restriction n,a\restriction n)\}$, that is, $A$ is recursively definable by a formula with only existential quantifiers ranging on members of $\omega^\omega$ or on $\omega$ and whose only parameter is $a$.<br />
<br />
The (non-relativized) '''lightface projective classes''', also known as ''analytical pointclasses'', are the special cases $\Sigma^1_n$, $\Pi^1_n$ and $\Delta^1_n$ of relativized lightface projective pointclasses where $a=\empty$. Let $\Sigma^0_1$ be the pointclass of all ''recursively enumerable'' sets, i.e. the sets $A$ such there exists a recursive relation $R$ such that $A=\{x\in\omega^\omega:\exists n\in\omega$ $R(x\restriction n)\}$, and $\Pi^0_1$ contain the completements of $\Sigma^0_1$ sets. Then the $\Sigma^1_1$ sets are precisely the projections of $\Pi^0_1$ sets.<br />
<br />
Given an arbitrary pointclass $\Gamma$, define $\neg\Gamma$ as the set of the complements of $\Gamma$'s elements, for example $\Pi^1_n(a)=\neg\Sigma^1_n(a)$. Also let $\Delta_\Gamma=\Gamma\cap\neg\Gamma$, for example $\Delta^1_n(a)=\Delta_{\Pi^1_n(a)}=\Delta_{\Sigma^1_n(a)}$.<br />
<br />
== Properties ==<br />
<br />
Every $\mathbf{\Sigma}^1_n$ set is $\Sigma^1_n(a)$ for some $a\in\omega^\omega$, in fact $\mathbf{\Sigma}^1_n=\bigcup_{a\in\omega^\omega}\Sigma^1_n(a)$. A similar statement holds for $\mathbf{\Pi}^1_n$ sets and $\mathbf{\Delta}^1_n$ sets. This means the boldface projective sets are precisely the set definable using only real and arithmetical quantifiers and real parameters.<br />
<br />
The following statements also holds when replacing relativized lightface pointclasses by their boldface counterparts:<br />
* If $A$ and $B$ are $\Sigma^1_n(a)$ relations, then so are $\exists x\in\omega^\omega$ $A$, $A\land B$, $A\lor B$, $\exists n\in\omega$ $A$ and $\forall n\in\omega$ $A$.<br />
* If $A$ and $B$ are $\Pi^1_n(a)$ relations, then so are $\forall x\in\omega^\omega$ $A$, $A\land B$, $A\lor B$, $\exists n\in\omega$ $A$ and $\forall n\in\omega$ $A$.<br />
* If $A$ is a $\Sigma^1_n(a)$ relation then $\neg A$ is a $\Pi^1_n(a)$ relation. If $A$ is $\Pi^1_n(a)$ then $\neg A$ is $\Sigma^1_n(a)$.<br />
* If $A$ is a $\Sigma^1_n(a)$ relation and $B$ is a $\Pi^1_n(a)$ relation, then $A\Rightarrow B$ is a $\Pi^1_n(a)$ relation.<br />
* If $A$ is a $\Pi^1_n(a)$ relation and $B$ is a $\Sigma^1_n(a)$ relation, then $A\Rightarrow B$ is a $\Sigma^1_n(a)$ relation.<br />
* If $A$ and $B$ are $\Delta^1_n(a)$, then so are $\neg A$, $A\land B$, $A\lor B$, $A\Rightarrow B$, $A\Leftrightarrow B$, $\exists n\in\omega$ $A$, $\forall n\in\omega$ $A$.<br />
* $\Delta^1_n(a)\subsetneq\Sigma^1_n(a)\subsetneq\Delta^1_{n+1}(a)$<br />
* $\Delta^1_n(a)\subsetneq\Pi^1_n(a)\subsetneq\Delta^1_{n+1}(a)$<br />
<br />
Sierpinski showed that every $\mathbf{\Sigma}^1_2$ set of reals is the union of $\aleph_1$ Borel (=$\mathbf{\Delta}^1_1$) sets. It follows that every $\mathbf{\Sigma}^1_2$ set of reals is either (at most) countable or has the cardinality of the continuum.<br />
<br />
''Shoenfield’s absoluteness theorem'' is the statement that every $\Sigma^1_2(a)$ or $\Pi^1_2(a)$ relation is absolute for every inner model of ZF+DC that contains $a$ (as an element). It follows that $\mathbf{\Sigma}^1_2$ and $\mathbf{\Pi}^1_2$ relations are absolute for $L$, and also that every $\Sigma^1_2(a)$ real (by taking $X=\omega$) is in $L[a]$, in particular every $\Sigma^1_2$ (or $\Pi^1_2$) real is constructible.<br />
The set of all constructible reals is $\Sigma^1_2$, and so is the canonical well-ordering $<_L$ of $L$. For $U$ a nonprincipal $\kappa$-complete [[ultrafilter]] on some [[measurable]] cardinal $\kappa$, then the collection of all sets of reals in $L[U]$ is $\Sigma^1_3$, and so is the canonical well-ordering $<_{L[U]}$ of $L[U]$. <br />
<br />
If [[zero sharp|$0^\#$]] exists then it is a $\Sigma^1_3$ real and the singleton $\{0^\#\}$ is a $\Pi^1_2$ set of reals. If for every real $a\in\omega^\omega$, the sharp $a^\#$ exists then every $\Sigma^1_3$ set of reals is the union of $\aleph_2$ Borel sets.<br />
<br />
== Regularity properties ==<br />
<br />
Let $A\subseteq(\omega^\omega)^k$ be a k-dimensional set of reals. We say that $A$ is ''null'' if it has [[:wikipedia:outer measure#Method I|outer measure]] 0. We say that $A$ is ''nowhere dense'' if its complement contains an open dense set, and that $A$ is ''meagre'' (or ''of first category'') if it is a countable union of nowhere dense set. Finally we say that $A$ is ''perfect'' if it has no isolated point.<br />
<br />
Then, we define the following ''regularity properties'':<br />
* $A$ is ''Lebesgue measurable'' if there exists a Borel set $B$ such that $A\Delta B$ is null.<br />
* $A$ has the ''Baire property'' if there exists an open set $B$ such that $A\Delta B$ is meagre.<br />
* $A$ has the ''perfect set property'' if it is either countable or has a perfect subset.<br />
Where $A\Delta B=(A\setminus B)\cup(B\setminus A)$ denotes symmetric difference. In ZFC there exists Lebesgue non-measurable sets without the Baire property nor the perfect set property, but it is interesting to see which projective sets have those regularity properties.<br />
<br />
In $L$ there is a $\Delta^1_2$ set of reals that is not Lebesgue measurable and has no perfect subset. Also there is a $\Pi^1_1$ set of reals without the perfect set property.<br />
<br />
If every $\Sigma^1_3$ set of reals is Lebesgue measurable then $\aleph_1$ is [[inaccessible]] in $L$.<br />
<br />
If $A$ is a $\Sigma^1_2(a)$ set of reals and contains a real that is not in $L[a]$ then $A$ has the perfect set property. Note that every uncountable set with the perfect set property has the cardinality of the continuum.<br />
<br />
The following statements are equivalent:<br />
* For every real $a$, $\aleph_1^{L[a]}$ is countable.<br />
* Every $\mathbf{\Pi}^1_1$ set has the perfect set property.<br />
* Every $\mathbf{\Sigma}^1_2$ set has the perfect set property.<br />
<br />
If $E$ is a $\mathbf{\Pi}^1_1$ equivalence relation on $\omega^\omega$ then either $E$ has at most $\aleph_0$ equivalence classes or there exists a perfect set of mutually inequivalent reals. If $E$ is a $\mathbf{\Sigma}^1_1$ equivalence relation on $\omega^\omega$ then either $E$ has at most $\aleph_1$ equivalence classes or there exists a perfect set of mutually inequivalent reals.<br />
<br />
=== Prewellordering, scale and uniformization properties ===<br />
<br />
A ''norm'' on a set $A$ is a function $\varphi:A\to Ord$ from $A$ to the ordinals. A ''prewellordering'' is a relation $\preceq$ that is like a well-ordering except we do not require it to be reflexive or antisymmetric. If $\preceq$ is a prewellordering then the $a\equiv b\iff(a\preceq b\land b\preceq a)$ is an equivalence relation, and $\preceq$ is a well-ordering of the equivalence classes of $\equiv$. If $\varphi$ is a norm then $a\preceq_\varphi b\iff\varphi(a)\leq\varphi(b)$ is a prewellordering.<br />
<br />
A pointclass $\Gamma$ has the ''prewellordering property'' if every set $A$ in $\Gamma$ has a $\Gamma$-norm: a norm $\varphi:A\to Ord$, a $\Gamma$ relation $P(x,y)$ and a $\neg\Gamma$ relation $Q(x,y)$ such that for every $y\in A$ and all $x$: $x\in A\land\varphi(x)\leq\varphi(y)\iff P(x,y)\iff Q(x,y)$. The pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the prewellordering property.<br />
<br />
A ''scale'' on $A$ is a sequence of norms $\{\varphi_n:n\in\omega\}$ such that for every sequence of points $\{x_i:i\in\omega\}$, if for every $n$ the sequence $\{\varphi_n(x_i):n\in\omega\}$ is eventually constant with value $\alpha_n$, then $(lim_{i\to\omega}$ $x_i)\in A$ and for every $n$, $\varphi_n(lim_{i\to\omega}$ $x_i)\leq\alpha_n$. A pointclass $\Gamma$ has the ''scale property'' if for every set $A$ in $\Gamma$, there exists a $\Gamma$-scale for $A$: a scale $\{\varphi_n:n\in\omega\}$, a $\Gamma$ relation $P(n,x,y)$ and a $\neg\Gamma$ relation $Q(x,y)$ such that for every $n$, every $y\in A$ and all $x$: $x\in A\land\varphi_n(x)\leq\varphi_n(y)\iff P(n,x,y)\iff Q(n,x,y)$. Again, the pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the scale property.<br />
<br />
A set $A\subseteq\omega^\omega\times\omega^\omega$ is ''uniformized'' by a function $F$ if $dom(F)=\{x\in\omega^\omega:\exists y\in\omega^\omega$ $(x,y)\in A\}$ and $(x,F(x))\in A$ for all $x\in dom(F)$. A pointclass $\Gamma$ has the ''uniformization'' property if every set in $\Gamma$ can be uniformized by a function in $\Gamma$. The pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the uniformization property.<br />
<br />
=== Reduction and separation properties ===<br />
<br />
For any four sets $A$, $B$, $A'$ and $B'$, $(A',B')$ ''reduces'' $(A,B)$ if $A'\subseteq A$, $B'\subseteq B$, $A'\cup B'=A\cup B$, but $A'\cap B'=\empty$. Thus $A'$ and $B'$ partition $A\cup B$. A pointclass $\Gamma$ has the ''reduction property'' if for every $A$, $B$ in $\Gamma$ there exists $A'$, $B'$ in $\Gamma$ such that $(A',B')$ reduces $(A,B)$. $\Pi^1_1(a)$ has the reduction property for every $a\in\omega^\omega$.<br />
<br />
A pointclass $\Gamma$ has the ''separation property'' if for any disjoint subsets $A$, $B$ of $(\omega^\omega)^k$ for some $k$, if $A$ and $B$ are in $\Gamma$ then there is a $C$ in $\Delta_\Gamma$ such that $A\subseteq C$ and $B\cap C=\empty$. $\Sigma^1_1(a)$ has the separation property for every $a\in\omega^\omega$.<br />
<br />
If $\Gamma$ has the reduction property, then $\neg\Gamma$ has the separation property but not the reduction property. It is impossible for $\Gamma$ to have both the reduction and the separation properties. Every pointclass with the prewellordering property has the reduction property. Thus it is impossible for both a pointclass and its complement to have the prewellordering or scale properties.<br />
<br />
== Projective determinacy ==<br />
<br />
''See also: [[axiom of determinacy]]''<br />
<br />
''Determinacy'' is a kind of regularity property. For every set of reals $A$, the game $G_A$ is the infinite game of perfect information of length $\omega$ where both players constructs a sequence (i.e. a real) by playing elements of $\omega$, one after the other, such that the first player's goal is to have the constructed real be in $A$, and the second player's goal is to have the constructed real be in $A$'s complement. $A$ is ''determined'' if the game $G_A$ is determined, i.e. one of the two players have a winning strategy for $G_A$.<br />
<br />
Given a pointclass $\Gamma$, ''$\Gamma$-determinacy'' is the statement "every $A\in\Gamma$ is determined". $\Gamma$-determinacy and $\neg\Gamma$-determinacy are always equivalent. $\omega^\omega$-determinacy is the ''axiom of determinacy'' and is implied false by the [[axiom of choice]]. The '''axiom of projective determinacy''' ($PD$) is precisely $(\bigcup_{n\in\omega}\mathbf{\Sigma}^1_n)$-determinacy. Given some class $M$ (e.g. $OD$, $L(\mathbb{R})$, ...), ''$M$-determinacy'' is an abbreviation for $(M\cap\mathcal{P}(\omega^\omega))$-determinacy. $L(\mathbb{R})$-determinacy notably follows from large cardinal axioms, in particular the existence of infinitely many [[Woodin]] cardinals with a [[measurable]] above them all.<br />
<br />
Martin showed that ZFC alone is sufficient to prove Borel determinacy (i.e. $\mathbf{\Delta}^1_1$-determinacy). However, for every $a\in\omega^\omega$, $\Sigma^1_1(a)$-determinacy is equivalent to "the sharp $a^\#$ exists", thus Borel determinacy is the best result possible in ZFC alone. Analytic (i.e. $\mathbf{\Sigma^1_1}$) determinacy follows from the existence of a measurable cardinal, or even just of a [[Ramsey]] cardinal. Stronger forms of projective determinacies requires considerably stronger large cardinal axioms: for every $n$, $\mathbf{\Delta}^1_{n+1}$-determinacy implies the existence of an inner model with $n$ Woodin cardinals.<br />
<br />
Note that for every $n$, $\mathbf{\Sigma}^1_n$-determinacy is equivalent to $\mathbf{\Pi}^1_n$-determinacy. Furthertmore, under $DC$ (the ''[[:wikipedia:axiom of dependent choice|axiom of dependent choice]]'') for every $n\in\omega$, $\mathbf{\Delta}^1_{2n}$-determinacy is equivalent to $\mathbf{\Sigma}^1_{2n}$-determinacy ($\mathbf{\Pi}^1_{2n}$-determinacy)<br />
<br />
Assume $\mathbf{\Sigma}^1_n$ (or $\mathbf{\Pi}^1_n$) determinacy and that the axiom of choice holds for ''countable'' sets of reals (which follows from $DC$). Then every $\mathbf{\Sigma}^1_{n+1}$ set of reals is Lebesgue measurable, has the Baire property and has the perfect set property.<br />
<br />
Assume projective determinacy; then the following pointclasses have the reduction, prewellordering, scale and uniformization properties, for every $a\in\omega^\omega$: $\Pi^1_1(a), \Sigma^1_2(a), \Pi^1_3(a), ..., \Pi^1_{2n+1}(a), \Sigma^1_{2n+2}(a), ...$ This is known as the ''periodicity theorem''. On the other hand, if $L[U]$ contains every real for some nonprincipal $\kappa$-complete [[ultrafilter]] $U$ on a measurable cardinal $\kappa$, then every $\Sigma^1_n(a)$ has the reduction and prewellordering properties for $n\geq 2$ and every $a\in\omega^\omega$.<br />
<br />
=== Projective ordinals ===<br />
<br />
For every pointclass $\Gamma$, define $\delta_\Gamma$ as the supremum of the length of $\Gamma$ prewellorderings of $\omega^\omega$. We then define the ''projective ordinals'' to be $\delta^1_n=\delta_{\mathbf{\Sigma}^1_n}=\delta_{\mathbf{\Pi}^1_n}$. It can be shown without $AD$ that $\delta^1_1=\omega_1$ and that $\delta^1_2\leq\omega_2$. Under $AD$, each projective ordinal is a regular cardinal and the sequence $\{\delta^1_n:n\in\omega\}$ is a strictly increasing sequence of measurable cardinals, also $\delta^1_2=\omega_2$, $\delta^1_3=\omega_{\omega+1}$ and $\delta^1_4=\omega_{\omega+2}$. In general, $\delta^1_{2n+2}\leq(\delta^1_{2n+1})^{+}$. Under $DC$ this becomes an equality, also every $\delta^1_{2n+1}$ is the successor of a cardinal of cofinality $\omega$.<br />
<br />
Define $E:\omega\to\omega_1$ by recursion the following way: $E(0)=1$, $E(n+1)=\omega^{E(n)}$ (ordinal exponentiation). Then, under $AD+DC$, one have $\delta^1_{2n+3}=\omega_{E(2n+1)+1}$, also every $\delta^1_{2n+3}$ has the strong partition property $\delta^1_{2n+3}\to(\delta^1_{2n+3})^{\delta^1_{2n+3}}_\alpha$ for every $\alpha<\delta^1_{2n+3}$.<br />
<br />
Let's say a set of reals $A$ is $\gamma$-Borel (for a cardinal $\gamma$) if it is in the smallest collection of sets containing all closed sets of $(\omega^\omega)^k$ that is closed under complementations and unions of less than $\gamma$ sets. If $\gamma$ is not a cardinal then $A$ is $\gamma$-Borel if it is $\gamma^{+}$-Borel where $\gamma^{+}$ is the smallest cardinal >$\gamma$. Note that a set is Borel if and only if it is $\aleph_1$-Borel.<br />
<br />
Assume $\mathbf{\Delta}^1_{2n}$-determinacy; then a set of reals $A$ is $\mathbf{\Delta}^1_{2n+1}$ if and only if it is $\delta^1_{2n+1}$-Borel. Now, assume $AD+DC$; then a set $A$ is $\mathbf{\Sigma}^1_{2n+2}$ if and only if it is the union of $\delta^1_{2n+1}$-many sets that are $\mathbf{\Delta}^1_{2n+1}$.<br />
<br />
=== Projective determinacy from large cardinals ===<br />
<br />
Woodin showed that $\mathbf{\Pi}^1_{n+1}$-determinacy follows from the existence of $n$ [[Woodin]] cardinals with a measurable above them all, and projective determinacy thus follows from the existence of infinitely many Woodin cardinals. He also showed that $\mathbf{\Pi}^1_2$-determinacy is equivalent to "for all $x\in\mathbb{R}$, there is a countable ordinal $\delta$ such that $\delta$ is a Woodin cardinal in some inner model of ZFC containing $x$, and that $\mathbf{\Delta}^1_2$-determinacy is equivalent to "for every $x\in\mathbb{R}$, there is an inner model M such that $x\in M$ and $M\models ZFC+$"there is a Woodin cardinal". <cite>KoellnerWoodin2010:LCFD</cite><br />
<br />
ZFC + (lightface) $\Delta^1_2$-determinacy implies that there many $x$ such that HOD$^{L[x]}$ is a model of ZFC+"$\omega_2^{L[x]}$ is a Woodin cardinal".<br />
Z$_2$+$\Delta^1_2$-determinacy is conjectured to be equiconsistent with ZFC+"Ord is Woodin", where "Ord is Woodin" is expressed as an axiom scheme and Z$_2$ is [[:wikipedia:second-order arithmetic|second-order arithmetic]].<br />
Z$_3$+$\Delta^1_2$-determinacy is provably equiconsistent with NBG+"Ord is Woodin" where NBG is [[:wikipedia:Von Neumann–Bernays–Gödel set theory|Von Neumann–Bernays–Gödel set theory]] and $Z_3$ is third-order arithmetic.<br />
<br />
Gitik and Schindler showed that, in ZF, if $\aleph_\omega$ is a strong limit cardinal and $2^{\aleph_\omega}>\aleph_{\omega_1}$, then the axiom of projective determinacy holds. Also, if there is a singular cardinal of uncountable cofinality such that the sets of the cardinals below it such that the GCH holds is both [[stationary]] and costationary, then again the axiom of projective determinacy holds. It is not known whether these two results extends to $L(\mathbb{R})$-determinacy. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
Foreman, Magidor and Schindler showed that if there exists infinitely many cardinals $\delta$ above the continuum such that both $\delta$ and $\delta^{+}$ have the [[tree property]], then the axiom of projective determinacy holds. This hypothesis was shown to be consistent relative to the existence of infinitely many [[supercompact]] cardinals by James Cummings and Foreman. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Projective&diff=2061Projective2017-11-11T19:58:18Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: Projective sets and determinacy}}<br />
We say that $\Gamma$ is a ''pointclass'' if it is a collection of subsets of a [[:wikipedia:Polish space|Polish space]]. <br />
The '''lightface and bolface projective hierarchies''' are hierarchies of pointclasses of some Polish space $X$ defined by repeated applications of projections and complementations from either recursively enumerable or closed sets respectively.<br />
<br />
''Most results in this article can be found in <cite>Jech2003:SetTheory</cite> and <cite>Kanamori2009:HigherInfinite</cite> unless indicated otherwise.''<br />
<br />
== Definitions ==<br />
<br />
The following definitions are made by taking $X=\omega^\omega$, the ''Baire space'', i.e. the set of all functions $f:\mathbb{N}\to\mathbb{N}$. We will identify its members with the corresponding real numbers under some fixed bijection between $\mathbb{R}$ and $\omega^\omega$. The definitions presented here can be easily extended to other Polish spaces than the Baire space.<br />
<br />
Let $\mathbf{\Sigma}^0_1$ be the pointclass that contains all open subsets of the Polish space $\omega^\omega$. Let $\mathbf{\Pi}^0_1$ be the pointclass containing the complements of the $\mathbf{\Sigma}^0_1$ sets.<br />
<br />
We define the '''boldface projective pointclasses''' $\mathbf{\Sigma}^1_n$, $\mathbf{\Pi}^1_n$ and $\mathbf{\Delta}^1_n$ the following way:<br />
# $\mathbf{\Sigma}^1_1$ contains all the images of $\mathbf{\Pi}^0_1$ sets by continuous functions; its members are called the ''analytic sets''.<br />
# Now, for all $n$, define $\mathbf{\Pi}^1_n$ to be the set of the complements of the $\mathbf{\Sigma}^1_n$ sets; the members of $\mathbf{\Pi}^1_1$ are called the ''coanalytic sets''.<br />
# For all $n$, define $\mathbf{\Sigma}^1_{n+1}$ to be the set of the images of $\mathbf{\Pi}^1_n$ sets by continuous functions.<br />
# Finally, let $\mathbf{\Delta}^1_n=\mathbf{\Sigma}^1_n\cap\mathbf{\Pi}^1_n$. The members of $\mathbf{\Delta}^1_1$ are the ''Borel'' sets.<br />
<br />
The '''relativized lightface projective pointclasses''' $\Sigma^1_n(a)$, $\Pi^1_n(a)$ and $\Delta^1_n(a)$ (for $a\in\omega^\omega$) are defined similarly except that $\Sigma^1_1(a)$ is defined as the set of all $A\subseteq\omega^\omega$ such that $A=\{x\in\omega^\omega:\exists y\in\omega^\omega$ $\exists n\in\omega$ $R(x\restriction n,y\restriction n,a\restriction n)\}$, that is, $A$ is recursively definable by a formula with only existential quantifiers ranging on members of $\omega^\omega$ or on $\omega$ and whose only parameter is $a$.<br />
<br />
The (non-relativized) '''lightface projective classes''', also known as ''analytical pointclasses'', are the special cases $\Sigma^1_n$, $\Pi^1_n$ and $\Delta^1_n$ of relativized lightface projective pointclasses where $a=\empty$. Let $\Sigma^0_1$ be the pointclass of all ''recursively enumerable'' sets, i.e. the sets $A$ such there exists a recursive relation $R$ such that $A=\{x\in\omega^\omega:\exists n\in\omega$ $R(x\restriction n)\}$, and $\Pi^0_1$ contain the completements of $\Sigma^0_1$ sets. Then the $\Sigma^1_1$ sets are precisely the projections of $\Pi^0_1$ sets.<br />
<br />
Given an arbitrary pointclass $\Gamma$, define $\neg\Gamma$ as the set of the complements of $\Gamma$'s elements, for example $\Pi^1_n(a)=\neg\Sigma^1_n(a)$. Also let $\Delta_\Gamma=\Gamma\cap\neg\Gamma$, for example $\Delta^1_n(a)=\Delta_{\Pi^1_n(a)}=\Delta_{\Sigma^1_n(a)}$.<br />
<br />
== Properties ==<br />
<br />
Every $\mathbf{\Sigma}^1_n$ set is $\Sigma^1_n(a)$ for some $a\in\omega^\omega$, in fact $\mathbf{\Sigma}^1_n=\bigcup_{a\in\omega^\omega}\Sigma^1_n(a)$. A similar statement holds for $\mathbf{\Pi}^1_n$ sets and $\mathbf{\Delta}^1_n$ sets. This means the boldface projective sets are precisely the set definable using only real and arithmetical quantifiers and real parameters.<br />
<br />
The following statements also holds when replacing relativized lightface pointclasses by their boldface counterparts:<br />
* If $A$ and $B$ are $\Sigma^1_n(a)$ relations, then so are $\exists x\in\omega^\omega$ $A$, $A\land B$, $A\lor B$, $\exists n\in\omega$ $A$ and $\forall n\in\omega$ $A$.<br />
* If $A$ and $B$ are $\Pi^1_n(a)$ relations, then so are $\forall x\in\omega^\omega$ $A$, $A\land B$, $A\lor B$, $\exists n\in\omega$ $A$ and $\forall n\in\omega$ $A$.<br />
* If $A$ is a $\Sigma^1_n(a)$ relation then $\neg A$ is a $\Pi^1_n(a)$ relation. If $A$ is $\Pi^1_n(a)$ then $\neg A$ is $\Sigma^1_n(a)$.<br />
* If $A$ is a $\Sigma^1_n(a)$ relation and $B$ is a $\Pi^1_n(a)$ relation, then $A\Rightarrow B$ is a $\Pi^1_n(a)$ relation.<br />
* If $A$ is a $\Pi^1_n(a)$ relation and $B$ is a $\Sigma^1_n(a)$ relation, then $A\Rightarrow B$ is a $\Sigma^1_n(a)$ relation.<br />
* If $A$ and $B$ are $\Delta^1_n(a)$, then so are $\neg A$, $A\land B$, $A\lor B$, $A\Rightarrow B$, $A\Leftrightarrow B$, $\exists n\in\omega$ $A$, $\forall n\in\omega$ $A$.<br />
* $\Delta^1_n(a)\subsetneq\Sigma^1_n(a)\subsetneq\Delta^1_{n+1}(a)$<br />
* $\Delta^1_n(a)\subsetneq\Pi^1_n(a)\subsetneq\Delta^1_{n+1}(a)$<br />
<br />
Sierpinski showed that every $\mathbf{\Sigma}^1_2$ set of reals is the union of $\aleph_1$ Borel (=$\mathbf{\Delta}^1_1$) sets. It follows that every $\mathbf{\Sigma}^1_2$ set of reals is either (at most) countable or has the cardinality of the continuum.<br />
<br />
''Shoenfield’s absoluteness theorem'' is the statement that every $\Sigma^1_2(a)$ or $\Pi^1_2(a)$ relation is absolute for every inner model of ZF+DC that contains $a$ (as an element). It follows that $\mathbf{\Sigma}^1_2$ and $\mathbf{\Pi}^1_2$ relations are absolute for $L$, and also that every $\Sigma^1_2(a)$ real (by taking $X=\omega$) is in $L[a]$, in particular every $\Sigma^1_2$ (or $\Pi^1_2$) real is constructible.<br />
The set of all constructible reals is $\Sigma^1_2$, and so is the canonical well-ordering $<_L$ of $L$. For $U$ a nonprincipal $\kappa$-complete [[ultrafilter]] on some [[measurable]] cardinal $\kappa$, then the collection of all sets of reals in $L[U]$ is $\Sigma^1_3$, and so is the canonical well-ordering $<_{L[U]}$ of $L[U]$. <br />
<br />
If [[zero sharp|$0^\#$]] exists then it is a $\Sigma^1_3$ real and the singleton $\{0^\#\}$ is a $\Pi^1_2$ set of reals. If for every real $a\in\omega^\omega$, the sharp $a^\#$ exists then every $\Sigma^1_3$ set of reals is the union of $\aleph_2$ Borel sets.<br />
<br />
== Regularity properties ==<br />
<br />
Let $A\subseteq(\omega^\omega)^k$ be a k-dimensional set of reals. We say that $A$ is ''null'' if it has [[:wikipedia:outer measure#Method I|outer measure]] 0. We say that $A$ is ''nowhere dense'' if its complement contains an open dense set, and that $A$ is ''meagre'' (or ''of first category'') if it is a countable union of nowhere dense set. Finally we say that $A$ is ''perfect'' if it has no isolated point.<br />
<br />
Then, we define the following ''regularity properties'':<br />
* $A$ is ''Lebesgue measurable'' if there exists a Borel set $B$ such that $A\Delta B$ is null.<br />
* $A$ has the ''Baire property'' if there exists an open set $B$ such that $A\Delta B$ is meagre.<br />
* $A$ has the ''perfect set property'' if it is either countable or has a perfect subset.<br />
Where $A\Delta B=(A\setminus B)\cup(B\setminus A)$ denotes symmetric difference. In ZFC there exists Lebesgue non-measurable sets without the Baire property nor the perfect set property, but it is interesting to see which projective sets have those regularity properties.<br />
<br />
In $L$ there is a $\Delta^1_2$ set of reals that is not Lebesgue measurable and has no perfect subset. Also there is a $\Pi^1_1$ set of reals without the perfect set property.<br />
<br />
If every $\Sigma^1_3$ set of reals is Lebesgue measurable then $\aleph_1$ is [[inaccessible]] in $L$.<br />
<br />
If $A$ is a $\Sigma^1_2(a)$ set of reals and contains a real that is not in $L[a]$ then $A$ has the perfect set property. Note that every uncountable set with the perfect set property has the cardinality of the continuum.<br />
<br />
The following statements are equivalent:<br />
* For every real $a$, $\aleph_1^{L[a]}$ is countable.<br />
* Every $\mathbf{\Pi}^1_1$ set has the perfect set property.<br />
* Every $\mathbf{\Sigma}^1_2$ set has the perfect set property.<br />
<br />
If $E$ is a $\mathbf{\Pi}^1_1$ equivalence relation on $\omega^\omega$ then either $E$ has at most $\aleph_0$ equivalence classes or there exists a perfect set of mutually inequivalent reals. If $E$ is a $\mathbf{\Sigma}^1_1$ equivalence relation on $\omega^\omega$ then either $E$ has at most $\aleph_1$ equivalence classes or there exists a perfect set of mutually inequivalent reals.<br />
<br />
=== Prewellordering, scale and uniformization properties ===<br />
<br />
A ''norm'' on a set $A$ is a function $\varphi:A\to Ord$ from $A$ to the ordinals. A ''prewellordering'' is a relation $\preceq$ that is like a well-ordering except we do not require it to be reflexive or antisymmetric. If $\preceq$ is a prewellordering then the $a\equiv b\iff(a\preceq b\land b\preceq a)$ is an equivalence relation, and $\preceq$ is a well-ordering of the equivalence classes of $\equiv$. If $\varphi$ is a norm then $a\preceq_\varphi b\iff\varphi(a)\leq\varphi(b)$ is a prewellordering.<br />
<br />
A pointclass $\Gamma$ has the ''prewellordering property'' if every set $A$ in $\Gamma$ has a $\Gamma$-norm: a norm $\varphi:A\to Ord$, a $\Gamma$ relation $P(x,y)$ and a $\neg\Gamma$ relation $Q(x,y)$ such that for every $y\in A$ and all $x$: $x\in A\land\varphi(x)\leq\varphi(y)\iff P(x,y)\iff Q(x,y)$. The pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the prewellordering property.<br />
<br />
A ''scale'' on $A$ is a sequence of norms $\{\varphi_n:n\in\omega\}$ such that for every sequence of points $\{x_i:i\in\omega\}$, if for every $n$ the sequence $\{\varphi_n(x_i):n\in\omega\}$ is eventually constant with value $\alpha_n$, then $(lim_{i\to\omega}$ $x_i)\in A$ and for every $n$, $\varphi_n(lim_{i\to\omega}$ $x_i)\leq\alpha_n$. A pointclass $\Gamma$ has the ''scale property'' if for every set $A$ in $\Gamma$, there exists a $\Gamma$-scale for $A$: a scale $\{\varphi_n:n\in\omega\}$, a $\Gamma$ relation $P(n,x,y)$ and a $\neg\Gamma$ relation $Q(x,y)$ such that for every $n$, every $y\in A$ and all $x$: $x\in A\land\varphi_n(x)\leq\varphi_n(y)\iff P(n,x,y)\iff Q(n,x,y)$. Again, the pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the scale property.<br />
<br />
A set $A\subseteq\omega^\omega\times\omega^\omega$ is ''uniformized'' by a function $F$ if $dom(F)=\{x\in\omega^\omega:\exists y\in\omega^\omega$ $(x,y)\in A\}$ and $(x,F(x))\in A$ for all $x\in dom(F)$. A pointclass $\Gamma$ has the ''uniformization'' property if every set in $\Gamma$ can be uniformized by a function in $\Gamma$. The pointclasses $\Sigma^1_2$ and, for all $a\in\omega^\omega$, $\Pi^1_1(a)$ have the uniformization property.<br />
<br />
=== Reduction and separation properties ===<br />
<br />
For any four sets $A$, $B$, $A'$ and $B'$, $(A',B')$ ''reduces'' $(A,B)$ if $A'\subseteq A$, $B'\subseteq B$, $A'\cup B'=A\cup B$, but $A'\cap B'=\empty$. Thus $A'$ and $B'$ partition $A\cup B$. A pointclass $\Gamma$ has the ''reduction property'' if for every $A$, $B$ in $\Gamma$ there exists $A'$, $B'$ in $\Gamma$ such that $(A',B')$ reduces $(A,B)$. $\Pi^1_1(a)$ has the reduction property for every $a\in\omega^\omega$.<br />
<br />
A pointclass $\Gamma$ has the ''separation property'' if for any disjoint subsets $A$, $B$ of $(\omega^\omega)^k$ for some $k$, if $A$ and $B$ are in $\Gamma$ then there is a $C$ in $\Delta_\Gamma$ such that $A\subseteq C$ and $B\cap C=\empty$. $\Sigma^1_1(a)$ has the separation property for every $a\in\omega^\omega$.<br />
<br />
If $\Gamma$ has the reduction property, then $\neg\Gamma$ has the separation property but not the reduction property. It is impossible for $\Gamma$ to have both the reduction and the separation properties. Every pointclass with the prewellordering property has the reduction property. Thus it is impossible for both a pointclass and its complement to have the prewellordering or scale properties.<br />
<br />
== Projective determinacy ==<br />
<br />
''See also: [[axiom of determinacy]]''<br />
<br />
''Determinacy'' is a kind of regularity property. For every set of reals $A$, the game $G_A$ is the infinite game of perfect information of length $\omega$ where both players constructs a sequence (i.e. a real) by playing elements of $\omega$, one after the other, such that the first player's goal is to have the constructed real be in $A$, and the second player's goal is to have the constructed real be in $A$'s complement. $A$ is ''determined'' if the game $G_A$ is determined, i.e. one of the two players have a winning strategy for $G_A$.<br />
<br />
Given a pointclass $\Gamma$, ''$\Gamma$-determinacy'' is the statement "every $A\in\Gamma$ is determined". $\Gamma$-determinacy and $\neg\Gamma$-determinacy are always equivalent. $\omega^\omega$-determinacy is the ''axiom of determinacy'' and is implied false by the [[axiom of choice]]. The '''axiom of projective determinacy''' ($PD$) is precisely $(\bigcup_{n\in\omega}\mathbf{\Sigma}^1_n)$-determinacy. Given some class $M$ (e.g. $OD$, $L(\mathbb{R})$, ...), ''$M$-determinacy'' is an abbreviation for $(M\cap\mathcal{P}(\omega^\omega))$-determinacy. $L(\mathbb{R})$-determinacy notably follows from large cardinal axioms, in particular the existence of infinitely many [[Woodin]] cardinals with a [[measurable]] above them all.<br />
<br />
Martin showed that ZFC alone is sufficient to prove Borel determinacy (i.e. $\mathbf{\Delta}^1_1$-determinacy). However, for every $a\in\omega^\omega$, $\Sigma^1_1(a)$-determinacy is equivalent to "the sharp $a^\#$ exists", thus Borel determinacy is the best result possible in ZFC alone. Analytic (i.e. $\mathbf{\Sigma^1_1}$) determinacy follows from the existence of a measurable cardinal, or even just of a [[Ramsey]] cardinal. Stronger forms of projective determinacies requires considerably stronger large cardinal axioms: for every $n$, $\mathbf{\Delta}^1_{n+1}$-determinacy implies the existence of an inner model with $n$ Woodin cardinals.<br />
<br />
Note that for every $n$, $\mathbf{\Sigma}^1_n$-determinacy is equivalent to $\mathbf{\Pi}^1_n$-determinacy. Furthertmore, under $DC$ (the ''[[:wikipedia:axiom of dependent choice|axiom of dependent choice]]'') for every $n\in\omega$, $\mathbf{\Delta}^1_{2n}$-determinacy is equivalent to $\mathbf{\Sigma}^1_{2n}$-determinacy ($\mathbf{\Pi}^1_{2n}$-determinacy)<br />
<br />
Assume $\mathbf{\Sigma}^1_n$ (or $\mathbf{\Pi}^1_n$) determinacy and that the axiom of choice holds for ''countable'' sets of reals (which follows from $DC$). Then every $\mathbf{\Sigma}^1_{n+1}$ set of reals is Lebesgue measurable, has the Baire property and has the perfect set property.<br />
<br />
Assume projective determinacy; then the following pointclasses have the reduction, prewellordering, scale and uniformization properties, for every $a\in\omega^\omega$: $\Pi^1_1(a), \Sigma^1_2(a), \Pi^1_3(a), ..., \Pi^1_{2n+1}(a), \Sigma^1_{2n+2}(a), ...$ This is known as the ''periodicity theorem''. On the other hand, if $L[U]$ contains every real for some nonprincipal $\kappa$-complete [[ultrafilter]] $U$ on a measurable cardinal $\kappa$, then every $\Sigma^1_n(a)$ has the reduction and prewellordering properties for $n\geq 2$ and every $a\in\omega^\omega$.<br />
<br />
=== Projective ordinals ===<br />
<br />
For every pointclass $\Gamma$, define $\delta_\Gamma$ as the supremum of the length of $\Gamma$ prewellorderings of $\omega^\omega$. We then define the ''projective ordinals'' to be $\delta^1_n=\delta_{\mathbf{\Sigma}^1_n}=\delta_{\mathbf{\Pi}^1_n}$. It can be shown without $AD$ that $\delta^1_1=\omega_1$ and that $\delta^1_2\leq\omega_2$. Under $AD$, each projective ordinal is a regular cardinal and the sequence $\{\delta^1_n:n\in\omega\}$ is a strictly increasing sequence of measurable cardinals, also $\delta^1_2=\omega_2$, $\delta^1_3=\omega_{\omega+1}$ and $\delta^1_4=\omega_{\omega+2}$. In general, $\delta^1_{2n+2}\leq(\delta^1_{2n+1})^{+}$. Under $DC$ this becomes an equality, also every $\delta^1_{2n+1}$ is the successor of a cardinal of cofinality $\omega$.<br />
<br />
Define $E:\omega\to\omega_1$ by recursion the following way: $E(0)=1$, $E(n+1)=\omega^{E(n)}$ (ordinal exponentiation). Then, under $AD+DC$, one have $\delta^1_{2n+3}=\omega_{E(2n+1)+1}$, also every $\delta^1_{2n+3}$ has the strong partition property $\delta^1_{2n+3}\to(\delta^1_{2n+3})^{\delta^1_{2n+3}}_\alpha$ for every $\alpha<\delta^1_{2n+3}$.<br />
<br />
Let's say a set of reals $A$ is $\gamma$-Borel (for a cardinal $\gamma$) if it is in the smallest collection of sets containing all closed sets of $(\omega^\omega)^k$ that is closed under complementations and unions of less than $\gamma$ sets. If $\gamma$ is not a cardinal then $A$ is $\gamma$-Borel if it is $\gamma^{+}$-Borel where $\gamma^{+}$ is the smallest cardinal >$\gamma$. Note that a set is Borel if and only if it is $\aleph_1$-Borel.<br />
<br />
Assume $\mathbf{\Delta}^1_{2n}$-determinacy; then a set of reals $A$ is $\mathbf{\Delta}^1_{2n+1}$ if and only if it is $\delta^1_{2n+1}$-Borel. Now, assume $AD+DC$; then a set $A$ is $\mathbf{\Sigma}^1_{2n+2}$ if and only if it is the union of $\delta^1_{2n+1}$-many sets that are $\mathbf{\Delta}^1_{2n+1}$.<br />
<br />
=== Projective determinacy from large cardinals ===<br />
<br />
Woodin showed that $\mathbf{\Pi}^1_{n+1}$-determinacy follows from the existence of $n$ [[Woodin]] cardinals with a measurable above them all, and projective determinacy thus follows from the existence of infinitely many Woodin cardinals. He also showed that $\mathbf{\Pi}^1_2$-determinacy is equivalent to "for all $x\in\mathbb{R}$, there is a countable ordinal $\delta$ such that $\delta$ is a Woodin cardinal in some inner model of ZFC containing $x$, and that $\mathbf{\Delta}^1_2$-determinacy is equivalent to "for every $x\in\mathbb{R}$, there is an inner model M such that $x\in M$ and $M\models ZFC+$"there is a Woodin cardinal". <cite>KoellnerWoodin2010:LCFD</cite><br />
<br />
ZFC+ (lightface) $\Delta^1_2$-determinacy implies that there many $x$ such that $HOD^{L[x]}\models ZFC+$"$\omega_2^{L[x]}$ is a Woodin cardinal".<br />
Z$_2$+$\Delta^1_2$-determinacy is conjectured to be equiconsistent with ZFC+"Ord is Woodin", where "Ord is Woodin" is expressed as an axiom scheme and Z$_2$ is [[:wikipedia:second-order arithmetic|second-order arithmetic]].<br />
Z$_3$+$\Delta^1_2$-determinacy is provably equiconsistent with $NBG+$"Ord is Woodin" where NBG is [[:wikipedia:Von Neumann–Bernays–Gödel set theory|Von Neumann–Bernays–Gödel set theory]] and $Z_3$ is third-order arithmetic.<br />
<br />
Gitik and Schindler showed that, in ZF, if $\aleph_\omega$ is a strong limit cardinal and $2^{\aleph_\omega}>\aleph_{\omega_1}$, then the axiom of projective determinacy holds. Also, if there is a singular cardinal of uncountable cofinality such that the sets of the cardinals below it such that the $GCH$ holds is both [[stationary]] and costationary, then again the axiom of projective determinacy holds. It is not known whether these two results extends to $L(\mathbb{R})$-determinacy. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
Foreman, Magidor and Schindler showed that if there exists infinitely many cardinals $\delta$ above the continuum such that both $\delta$ and $\delta^{+}$ have the [[tree property]], then the axiom of projective determinacy holds. This hypothesis was shown to be consistent relative to the existence of infinitely many [[supercompact]] cardinals by James Cummings and Foreman. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Axiom_of_determinacy&diff=2060Axiom of determinacy2017-11-11T19:55:43Z<p>Wabb2t: </p>
<hr />
<div>The '''axiom of determinacy''' is the assertion that a certain type of two-player games of perfect information (i.e. games in which the players alternate moves which are known to both players, and the outcome of the game depends only on this list of moves, and not on chance or other external factors) are ''determined'', that is, there is an "optimal strategy" that allows one player to win regardless of the other player's strategy. That strategy is called a ''winning strategy'' for that player.<br />
<br />
The axiom of determinacy is incompatible with the [[axiom of choice]]. More precisely, it is incompatible with the existence of a well-ordering of the reals. The AD is, however, not known to be inconsistent with [[ZFC|ZF]] set theory. AD is furthermore a very powerful axiom, as ZF+AD implies the consistency of ZF, ZF+Con(ZFC), and much more - it is in fact close of being a large cardinal axiom, as Woodin proved that it was equiconsistent with the existence of infinitely many [[Woodin]] cardinals. <cite>KoellnerWoodin2010:LCFD</cite><br />
<br />
It follows from large cardinal axioms (in particular from the existence of infinitely many Woodins with a [[measurable]] above them all <cite>KoellnerWoodin2010:LCFD</cite>) that the AD is true in $L(\mathbb{R})$, the [[constructible universe]] obtained by starting with the transitive closure of the set of all reals (i.e. $L_0(\mathbb{R})=TC(\{\mathbb{R}\})$). This assertion, generally refered to as $L(\mathbb{R})$-determinacy, AD$^{L(\mathbb{R})}$ <cite>KoellnerWoodin2010:LCFD</cite> or ''quasi-projective determinacy'' <cite>Maddy88:BelAxiomsII</cite> is not known to be inconsistent with ZFC. $L(\mathbb{R})$-determinacy is furthermore equiconsistent with AD (in $V$). A particular case of this is the [[axiom of projective determinacy]] which states that every [[projective]] set is determined, projectivity being a weak form of definability (more precisely definability in second-order arithmethic).<br />
<br />
== Type of games that are determined ==<br />
<br />
Given a set $S$ of infinite sequences of order-type (length) $\omega$ (i.e, a subset of the Baire Space $\omega^{\omega}$), the ''payoff'' set, the game begins as such: Player I says a natural number $n_0$, then Player II says a natural number $n_1$, and so on, until a sequence of order-type $\omega$ is constructed. At this point, a natural number $n_i$ has been given for every natural number $i$. Player I wins if $(n_0,n_1,n_2...)\in S$, Player II wins otherwise. Since $\omega^\omega$ and the set $\mathbb{R}$ of the real numbers are in bijection with the other, we shall often identify the elements of $\omega^\omega$ as the ''real numbers'', like if $\omega^\omega$ and $\mathbb{R}$ were equal. Thus the game considered here produces a real number.<br />
<br />
A ''strategy'' for player I (resp. player II) is a function $\Sigma$ with domain the set of sequences of integers of even (odd) length such that for each $a\in dom(\Sigma)$, $\Sigma(a)\in\omega$. A run of the game (partial or complete) is said to be according to a strategy $\Sigma$ for player player I (player II) if every initial segment of the run of odd (nonzero even) length is of the form $a\frown⟨\Sigma(a)⟩$ for some sequence $a$. A strategy $\Sigma$ for player player I (player II) is a winning strategy if every complete run of the game according to $\Sigma$ is in (out of) $S$. We say that a set $S\subset\omega^\omega$ is determined if there exists a winning strategy for one of the players<br />
<br />
The '''axiom of determinacy''' states that every payoff set $S\subset\omega^\omega$ is determined <cite>Larson2010:HistoryDeterminacy</cite>. It is possible to show that every finite or countable payoff set is determined, so this equivalent to the assertion that every uncountable payoff set is determined.<br />
<br />
== Refuting the axiom of determinacy from a well-ordering of the reals ==<br />
<br />
As stated above, the axiom of determinacy is not compatible with the axiom of choice, that is, within ZFC we can prove that axiom of determinacy fails. We outline a construction of an undetermined game starting from a well-ordering of continuum.<br />
<br />
A strategy for either player is a function with countable domain (a subset of the set of all finite sequences of integers) to $\omega$, so there are $2^{\aleph_0}$ many strategies for player I and $2^{\aleph_0}$ continuum many strategies for player II. Let $\{s^{I}_\alpha:\alpha<2^{\aleph_0}\}, \{s^{II}_\alpha:\alpha<2^{\aleph_0}\}$ be enumerations of strategies for the respective players. We shall now construct, by transfinite recursion, two disjoint sets of sequences $\{a_\alpha:\alpha<2^{\aleph_0}\}, \{b_\alpha:\alpha<2^{\aleph_0}\}\subseteq\omega^\omega$ such that $\{a_\alpha:\alpha<2^{\aleph_0}\}$ is not determined.<br />
<br />
Suppose that, for some $\beta<2^{\aleph_0}$, $\{a_\alpha:\alpha<\beta\},\{b_\alpha:\alpha<\beta\}$ have already been constructed. Take strategy $s^I_\beta$. There are continuum many possible plays according to this strategy (since player II can play in arbitrary way at any of their turns), so not all of them can be already contained in $\{a_\alpha:\alpha<\beta\}$ (which has cardinality $|\beta|<2^{\aleph_0}$). Therefore, using well-ordering of continuum, we can pick one of these plays and define it to be $b_\beta$. Similarly, we can pick $a_\beta$ according to strategy $s^{II}_\beta$ which is not already in $\{b_\alpha:\alpha\leq\beta\}$. This way the sets $\{a_\alpha:\alpha<2^{\aleph_0}\},\{b_\alpha:\alpha<2^{\aleph_0}\}$ are clearly disjoint.<br />
<br />
Letting $A=\{a_\alpha:\alpha<2^{\aleph_0}\}$, we now claim the game with payoff set $A$ is undetermined. Indeed, suppose player I has a winning strategy. This must be one of the strategies $s^I_\beta$. By construction, player II can arrange the play so that the resulting play is $b_\beta$ (since we have chosen it so that it's consistent with strategy $b_\beta$), which is not an element of $A$, contradicting the assumption that $s^I_\beta$ is a winning strategy. Analogously, for any strategy $s^{II}_\beta$ for player II, player I can force the play to be $a_\beta\in A$. Therefore no strategy for either player is a winning strategy and it follows that the game is undetermined.<br />
<br />
== Other known limitations of determinacy ==<br />
<br />
Assuming the axiom of choice there is a non-determined game of length $\omega$. However, choice isn't known to contradict the determinacy of all ''definable'' games of length $\omega$.<br />
<br />
With or without assuming choice, there is a non-determined game of length $\omega_1$ and a a non-determined definable game of length $\omega_1+\omega$. There is also a non-determined game of length $\omega$ with moves in $\omega_1$ (i.e. the payoff sets are subsets of $\omega_1^\omega$ instead of subsets of $\omega^\omega$. There is a non-determined game of length $\omega$ with moves in $\mathcal{P}(\mathbb{R})$, and using choice one can show there is such a game that is definable. [http://mathoverflow.net/questions/271507/limitations-of-determinacy-hypotheses-in-zfc]<br />
<br />
Definable games of length $\omega$ with moves in $\mathbb{R}$ are provably determined from large cardinal axioms. Determinacy of such games that are projective follows from the existence of sufficiently many Woodin cardinals.<br />
<br />
By a result of Woodin, if there is an iterable model of ZFC with a countable (in $V$) Woodin cardinal which is a limit of Woodin cardinals, then it is consistent (even with choice) that all ordinal-definable games of length $\omega_1$ are determined. This is only a consistency result, not a proof of "all ordinal-definable games of length $\omega_1$ are determined".<br />
<br />
== Implications of the axiom of determinacy ==<br />
<br />
Assume ZF+AD. Most of the following results can be found in <cite>Kanamori2009:HigherInfinite</cite>, in <cite>Larson2010:HistoryDeterminacy</cite> or in [http://mathoverflow.net/questions/129036/counterintuitive-consequences-of-the-axiom-of-determinacy]:<br />
<br />
* The [[:wikipedia:axiom of countable choice|axiom of countable choice]] restrained to countable sets of reals is true.<br />
* In $L(\mathbb{R})$ the [[:wikipedia:axiom of dependent choice|axiom of dependent choice]] is true.<br />
* The reals cannot be well-ordered. Thus the full [[axiom of choice]] fails.<br />
* Every set of reals is [[:wikipedia:Lebesgue measurable|Lebesgue measurable]]. Thus the [[:wikipedia:Banach-Tarski paradox|Banach-Tarski paradox]] fails.<br />
** It follows by a theorem of Raisonnier that $\omega_1\not\leq 2^{\aleph_0}$ (yet $\omega_1\not\geq2^{\aleph_0}$).<br />
** Furthermore, it implies $2^{\aleph_0}$ can be partitioned in more than $2^{\aleph_0}$ many pairwise disjoint nonempty subsets.<br />
* Every set of reals has the [[:wikipedia:Baire property|Baire property]].<br />
* Every set of reals is either [[countable]] or has a [[:wikipedia:perfect set|perfect subset]].<br />
** Thus a form of the [[continuum hypothesis]] holds, i.e. every set of reals is either countable or has cardinality $2^{\aleph_0}$.<br />
** Other forms of CH however fail, in particular $2^{\aleph_0}\neq\aleph_1$.<br />
* There are no free [[filter|ultrafilters]] on $\omega$. Every ultrafilter on $\omega$ is principal. Thus every ultrafilter is countably complete ($\aleph_1$-complete).<br />
* $\omega_1$, $\omega_2$, and $\omega_{\omega+1}$ and $\omega_{\omega+2}$ are all [[measurable]] cardinals.<br />
** The [[club]] filter on $\omega_1$ is an ultrafilter. Every subset of $\omega_1$ either contains a club or is disjoint from one.<br />
** The club filter on $\omega_2$ restrained to sets of [[cofinality]] $\omega$ is $\omega_2$-complete.<br />
* $\omega_n$ is singular for every $n>2$ and has cofinality $\omega_2$ and is [[Jonsson]], also $\aleph_\omega$ is [[Rowbottom]].<br />
* [[zero sharp|$0^{\#}$]] exists, thus the [[:wikipedia:axiom of constructibility|axiom of constructibility]] (V=L) fails.<br />
** In fact, $x^{\#}$ exists for every $x\in\mathbb{R}$ (thus $V\neq L[x]$).<br />
* The strong [[partition property]], $\omega_1\rightarrow(\omega_1)^{\omega_1}_2$, holds. In fact, $\omega_1\rightarrow(\omega_1)^{\omega_1}_{2^{\aleph_0}}$ and $\omega_1\rightarrow(\omega_1)^{\omega_1}_\alpha$ for every $\alpha<\omega_1$.<br />
* If there is a surjection $\mathbb{R}\to\alpha$, then there is surjection $\mathbb{R}\to\mathcal{P}(\alpha)$ (this is ''Moschovakis' coding lemma'').<br />
* [[:wikipedia:Hall's marriage theorem|Hall's marriage theorem]] fails for infinite graphs. For example there is there is a 2-regular bipartite graph on $\mathbb{R}$ with no perfect matching.<br />
* There is no [[:wikipedia:Basis (linear algebra)#Related_notions|Hamel basis]] of $\mathbb{R}$ over $\mathbb{Q}$.<br />
<br />
Let $\Theta$ be the supremum of the ordinals that $\mathbb{R}$ can be mapped onto. Under $AC$ this is just $(2^{\aleph_0})^{+}$ but under AD it is a limit cardinal, in fact an aleph fixed point, and DC implies it has uncountable cofinality. In $L(\mathbb{R})$ it is also regular and thus [[weakly inaccessible]]. It is conjectured that under AD the cofinality function is nondecreasing on singular cardinals below $\Theta$.<br />
<br />
== Determinacy of $L(\mathbb{R})$ ==<br />
<br />
''See also: [[Constructible universe]]''<br />
<br />
Recall that a formula $\varphi$ is $\Delta_0$ if and only if it only contains bounded quantifiers (i.e. $(\forall x\in y)$ and $(\exists x\in y)$). Let $def(X)=\{Y\subset X : Y$ is first-order definable by a $\Delta_0$ formula with parameters only from $X\cup\{X\}\}$. Then let:<br />
*$L_0(X)=TC(\{X\})$<br />
*$L_{\alpha+1}(X)=def(L_\alpha(X))$<br />
*$L_\lambda(X)=\bigcup_{\alpha<\lambda}L_\alpha(X)$ for limit $\lambda$<br />
*$L(X)=\bigcup_{\alpha\in Ord}L_\alpha(X)$<br />
where $TC({X})$ is the smallest transitive set containing $X$, the elements of $X$, the elements of the elements of $X$, and so on. $L(X)$is always a model of ZF, but not necessarily of the axiom of choice.<br />
<br />
$L(X,Y)$ is used as a shortcut for $L(\{X,Y\})$. $L(X,\mathbb{R})$ with $X\subset\mathbb{R}$ is different from $L(\mathbb{R})$ whenever $X$ is not constructible from the reals, i.e. $X\not\in L(\mathbb{R})$ (if any such set exists; it is consistent with ZF+AD that they do not). <br />
<br />
$L(\mathbb{R})$-determinacy, also known as AD$^{L(\mathbb{R})}$ <cite>KoellnerWoodin2010:LCFD</cite> or ''quasi-projective determinacy'' <cite>Maddy88:BelAxiomsII</cite> is the assertion that every set of reals in $L(\mathbb{R})$ is determined. Equivalently, "$L(\mathbb{R})$ is a model of ZF+AD".<br />
<br />
$L(\mathbb{R})$-determinacy appears to be a very "natural" statement in that, empirically, every natural extension of ZFC (i.e. not made specifically to contradict this) that is not proved consistent by AD seems to imply $L(\mathbb{R})$-determinacy or some weaker form of determinacy. <cite>Larson2010:HistoryDeterminacy</cite> This is often considered to be an argument toward the "truth" of $L(\mathbb{R})$-determinacy.<br />
<br />
Assuming ZF+DC, in $L(\mathbb{R})$ AD follows from three of its consequences: <cite>Larson2010:HistoryDeterminacy</cite><br />
# Every set of reals is Lebesgue measurable.<br />
# Every set of reals has the Baire property. <br />
# Every $\Sigma^1_2$ set of reals can be uniformized.<br />
<br />
In $L(\mathbb{R})$, the axiom of determinacy is equivalent to the axiom of Turing determinacy <cite>Larson2010:HistoryDeterminacy</cite>, i.e. the assertion that payoff sets closed under [[:wikipedia:Turing equivalence|Turing equivalence]] are determined.<br />
<br />
Busche and Schindler showed that, if there is a model of ZF in wich every uncountable cardinal is singular (thus has cofinality $\aleph_0$), then the axiom of determinacy holds in the $L(\mathbb{R})$ of some forcing extension of $HOD$ <cite>Larson2010:HistoryDeterminacy</cite>. This notably follows from the existence of a proper class of [[strongly compact]] cardinals. <br />
<br />
Assume that there is $\omega_1$-dense ideal over $\omega_1$; then $L(\mathbb{R})$-determinacy holds. <cite>Kanamori2009:HigherInfinite</cite> This result is due to Woodin.<br />
<br />
The following holds in $L(\mathbb{R})$ assuming $L(\mathbb{R})$-determinacy: <cite>KoellnerWoodin2010:LCFD</cite><cite>JacksonKetchersidSchlutzenbergWoodin:DeterminacyJonsson</cite> <br />
* Every uncountable cardinal $<\Theta$ is [[Jonsson|Jónsson]], also if it is regular or has cofinality $\omega$ then it is [[Rowbottom]].<br />
* Every regular cardinal $<\Theta$ is [[measurable]] (note that $2^{\aleph_0}\not\leq\Theta$), also $\Theta$ is a limit of measurable cardinals.<br />
* $\Theta$ is weakly $\Theta$-[[Mahlo]] (and thus weakly $\Theta$-inaccessible), but it is not [[weakly compact]].<br />
* $\omega_1$ is <$\Theta$-[[supercompact]], i.e. it is $\gamma$-supercompact for all $\gamma<\Theta$.<br />
* $\Theta$ is Woodin in the model $HOD^{L(\mathbb{R})}$.<br />
<br />
== Axiom of projective determinacy ==<br />
<br />
''Main article: [[Projective#Projective determinacy|Projective determinacy]]''<br />
<br />
== Axiom of real determinacy ==<br />
<br />
The '''axiom of real determinacy''' (AD$_\mathbb{R}$) is the assertion that if payoff sets contains real numbers instead of natural numbers, then every payoff set is still determined. This is strictly stronger than AD, and ZF+AD$_\mathbb{R}$ proves ZF+AD consistent.<br />
<br />
AD$_\mathbb{R}$ is equivalent (over ZF) to AD plus the [[:wikipedia:Uniformization (set theory)|axiom of uniformization]] (which is false in $L(\mathbb{R})$). AD$_\mathbb{R}$ is also equivalent to determinacy for games of length $\omega^2$. In fact, AD_$\mathbb{R}$ is equivalent to the assertion that every game of bounded countable length is determined. It is however possible to show (in ZF) that there are non-determined games of length $\aleph_1$.<br />
<br />
Solovay showed that ZF+AD$_\mathbb{R}$+"$\Theta$ has uncountable cofinality" (which follows from ZF+AD$_\mathbb{R}$+DC) proves ZF+AD$_\mathbb{R}$ consistent; it is therefore consistent with ZF+AD$_\mathbb{R}$ that $\Theta$ has cofinality $\omega$ and that DC is false. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
Steel showed that under AD$_\mathbb{R}$, in a forcing extension there is a proper class model of ZFC in which there exists a cardinal $\delta$ of cofinality $\aleph_0$ which is a limit of Woodin cardinals and <$\delta$-strong cardinals. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
Under AD_$\mathbb{R}$, $\omega_1$ is <$\Theta$-supercompact, i.e. for every ordinal $\gamma<\Theta$ there is a normal fine ultrafilter on the set of all subsets of $\gamma$ of size $\aleph_1$. AD suffices for this result to hold in $L(\mathbb{R})$, but is not known to suffice for it to hold in $V$. <cite>Larson2010:HistoryDeterminacy</cite><br />
<br />
A set $\Gamma\subset\mathcal{P}(\mathbb{R})$ is a ''Wadge initial segment'' of $\mathcal{P}(\mathbb{R})$ if for every $X\in\Gamma$, if $Y\leq_W X$ (i.e. $Y$ is [[:wikipedia:Wadge hierarchy|Wadge reducible]] to $X$) then $Y\in\Gamma$. Under suitable large cardinal assumptions, there exists a Wadge initial segment $\Gamma\subset\mathcal{P}(\mathbb{R})$ such that $L(\Gamma,\mathbb{R})$ is a model of "AD$^{+}$, AD$_\mathbb{R}$ and $\Gamma=\mathcal{P}(\mathbb{R})$" (see [[:wikipedia:AD+|AD+]]). Furthermore, whenever $\mathcal{M}$ is an inner model such that $\mathbb{R}\subset\mathcal{M}$ and $\mathcal{M}$ is a model of "AD$^{+}$ and AD$_\mathbb{R}$", one has $\Gamma\subset\mathcal{M}$. ''(see the 'Read more' section)''<br />
<br />
== Consistency strength of determinacy hypotheses ==<br />
<br />
The following theories are equiconsistent: <cite>Kanamori2009:HigherInfinite</cite><cite>TrangWilson2016:DetFromStrongCompactness</cite><br />
* ZF+AD<br />
* ZF+AD+DC<br />
* ZFC+AD$^L(\mathbb{R})$<br />
* ZFC+AD$^OD(\mathbb{R})$<br />
* ZFC+"the non-stationary ideal over $\omega_1$ is $\omega_1$-dense"<br />
* ZFC+"there exists infinitely many [[Woodin]] cardinals"<br />
* ZF+DC+"$\omega_1$ is $\mathcal{P}(\omega_1)$-[[strongly compact]]"<br />
* ZF+DC+"$\omega_1$ is $\mathbb{R}$-strongly compact and $\Theta>\omega_2$"<br />
* ZF+DC+"$\omega_1$ is $\mathbb{R}$-strongly compact and $\omega_2$-strongly compact"<br />
* ZF+DC+"$\omega_1$ is $\mathbb{R}$-strongly compact and Jensens's square principle fails for $\omega_1$"<br />
Where DC is the [[:wikipedia:axiom of dependent choice|axiom of dependent choice]] and $\omega_1$ being $X$-strongly compact means that there exists a [[filter|fine measure]] on the set of all subsets of $X$ of cardinality $\aleph_1$.<br />
<br />
[[Projective determinacy]] is a little weaker: it is equiconsistent with ZFC plus, for all n, an axiom saying "there are n Woodin cardinals". Since ZFC can only use finitely many of its axioms, this axiom schema does not allow ZFC to prove that there exists infinitely many Woodins, despite making it able to prove every particular instance of "there exists at least n Woodin cardinals".<br />
<br />
Koellner annd Woodin showed that the following theories are also equiconsistent: <cite>KoellnerWoodin2010:LCFD</cite><br />
* ZFC+$\Delta^1_2$-determinacy<br />
* ZFC+OD-determinacy<br />
* ZFC+"there exists a Woodin cardinal"<br />
<br />
And so are Z$_3$+lightface $\Delta^1_2$-determinacy and MK+"Ord is Woodin" where Z$_3$ is ''third-order arithmetic'' and MK is [[Morse-Kelley set theory]]. It is also conjectured that Z$_2$+$\Delta^1_2$-determinacy and ZFC+"Ord is Woodin" are equiconsistent, where Z$_2$ is [[:wikipedia:second-order arithmetic|second-order arithmetic]] and "Ord is Woodin" is expressed as an axiom scheme.<br />
<br />
Finally, Trang and Wilson proved that the following theories are equiconsistent: <cite>TrangWilson2016:DetFromStrongCompactness</cite><br />
* ZF+DC+AD$_\mathbb{R}$<br />
* ZF+DC+"$\omega_1$ is $\mathcal{P}(\mathbb{R})$-strongly compact"<br />
* ZF+DC+"$\omega_1$ is $\mathbb{R}$-strongly compact and $\Theta$ is singular"<br />
* ZF+DC+"$\omega_1$ is $\mathbb{R}$-strongly compact and $\Theta$-strongly compact"<br />
As are the following theories:<br />
* ZF+AD$_\mathbb{R}$<br />
* ZF+DC$_{\mathcal{P}(\omega_1)}$+"$\omega_1$ is $\mathbb{R}$-strongly compact and $\Theta$ is singular"<br />
<br />
== Read more ==<br />
<br />
* ''"Is there a natural inner model of AD$_\mathbb{R}$?"'' [http://mathoverflow.net/questions/269241/is-there-a-natural-inner-model-of-ad-mathbbr/269690]<br />
<br />
* ''"Limitations of determinacy hypotheses in ZFC"'' [http://mathoverflow.net/questions/271507/limitations-of-determinacy-hypotheses-in-zfc]<br />
<br />
* ''"Counterintuitive consequences of the Axiom of Determinacy?"'' [https://mathoverflow.net/questions/129036/counterintuitive-consequences-of-the-axiom-of-determinacy]<br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Positive_set_theory&diff=2059Positive set theory2017-11-11T19:40:37Z<p>Wabb2t: </p>
<hr />
<div>''Positive set theory'' is the name of a certain group of axiomatic set theories originally created as an example of a (nonstandard) set theories in which the axiom of foundation fails (e.g. there exists $x$ such that $x\in x$). <cite>FortiHinnion89:ConsitencyProblemPositiveComp</cite> Those theories are based on a weakening of the (inconsistent) ''comprehension axiom'' of [[naive set theory]] (which asserts that every formula $\phi(x)$ defines a set that contains all $x$ such that $\phi(x)$) by restraining the formulas used to a smaller class of formulas called ''positive'' formulas.<br />
<br />
While most positive set theories are weaker than [[ZFC]] (and usually mutually interpretable with [[:wikipedia:second-order arithmetic|second-order arithmetic]]), one of them, GPK$^+_\infty$ turns out to be very powerful, being mutually interpretable with [[Morse-Kelley set theory]] plus an axiom asserting that the class of all [[ordinal|ordinals]] is [[weakly compact]]. <cite>Esser96:InterpretationZFCandMKinPositiveTheory</cite><br />
<br />
== Positive formulas ==<br />
<br />
In the first-order language $\{=,\in\}$, we define a ''BPF formula'' (bounded positive formula) the following way <cite>Esser96:InterpretationZFCandMKinPositiveTheory</cite>:<br />
For every variable $x$, $y$ and BPF formulas $\varphi$, $\psi$,<br />
* $x=y$ and $x\in y$ are BPF.<br />
* $\varphi\land\psi$, $\varphi\lor\psi$, $\exists x\varphi$ and $(\forall x\in y)\varphi$ are BPF.<br />
<br />
A formula is then a ''GPF formula'' (generalized positive formula) if it is a BPF formula or if it is of the form $\forall x(\theta(x)\Rightarrow\varphi)$ with $\theta(x)$ a GPF formula with exactly one free variable $x$ and no parameter and $\varphi$ is a GPF formula (possibly with parameters). <cite>Esser96:GPKAFA</cite><br />
<br />
== GPK positive set theories ==<br />
<br />
The positive set theory GPK consists of the following axioms:<br />
* '''Empty set''': $\exists x\forall y(y\not\in x)$.<br />
* '''Extensionality''': $\forall x\forall y(x=y\Leftrightarrow\forall z(z\in x\Leftrightarrow z\in y))$.<br />
* '''GPF comprehension''': the universal closure of $\exists x\forall y(y\in x\Leftrightarrow\varphi)$ for every GPF formula $\varphi$ (with parameters) in which $x$ does not occur.<br />
The empty set axiom is necessary, as without it the theory would hold in the trivial model which has only one element satisfying $x=\{x\}$. Note that, while GPK do proves the existence of a set such that $x\in x$, Olivier Esser proved that it refutes the [[:wikipedia:anti-foundation axiom|anti-foundation axiom]] (AFA). <cite>Esser96:GPKAFA</cite><br />
<br />
The theory GPK$^+$ is obtained by adding the following axiom:<br />
* '''Closure''': the universal closure of $\exists x(\forall z(\varphi(z)\Rightarrow z\in x)\land\forall y(\forall w(\varphi(z)\Rightarrow z\in y)\Rightarrow y\subset x))$ for every formula $\varphi(z)$ (not necessarily BPF or GPF) with a free variable $z$ (and possibly parameters) such that $x$ does not occur in $\varphi$.<br />
This axiom scheme asserts that for any (possibly proper) class $C=\{x|\varphi(x)\}$ there is a smallest set $X$ containing $C$, i.e. $C\subset X$ and for all sets $Y$ such that $C\subset Y$, one has $X\subset Y$. <cite>Esser99:ConsistencyPositiveTheory</cite><br />
<br />
Note that replacing GPF comprehension in GPK$^+$ by BPF comprehension does not make the theory any weaker: BPF comprehension plus Closure implies GPF comprehension.<br />
<br />
Both GPK and GPK$^+$ are consistent relative to ZFC, in fact mutually interpretable with second-order arithmetic. However a much stronger theory, '''GPK$^+_\infty$''', is obtained by adding the following axiom:<br />
* '''Infinity''': the von Neumann ordinal $\omega$ is a set.<br />
By "von Neumann ordinal" we mean the usual definition of ordinals as well-ordered-by-inclusion sets containing all the smaller ordinals. Here $\omega$ is the set of all finite ordinals (the natural numbers). The point of this axiom is not implying the existence of an infinite set; the ''class'' $\omega$ exists, so it has a set closure which is certainely infinite. This set closure happens to satisfy the usual axiom of infinity of ZFC (i.e. it contains 0 and the successor of all its members) but in GPK$^+$ this is not enough to deduce that $\omega$ itself is a set (an improper class).<br />
<br />
Olivier Esser showed that GPK$^+_\infty$ can not only interpret ZFC (and prove it consistent), but is in fact mutually interpretable with a ''much'' stronger set theory, namely, Morse-Kelley set theory with an axiom asserting that the (proper) class of all ordinals is [[weakly compact]]. This theory is powerful enough to prove, for instance, that there exists a proper class of [[Mahlo|hyper-Mahlo]] cardinals. <cite>Esser96:InterpretationZFCandMKinPositiveTheory</cite><br />
<br />
== As a topological set theory ==<br />
''To be expanded.''<br />
== The axiom of choice and GPK set theories ==<br />
''To be expanded. <cite>Esser2000:InconsistencyACwithGPK</cite><cite>FortiHinnion89:ConsitencyProblemPositiveComp</cite>''<br />
== Other positive set theories and the inconsistency of the axiom of extensionality ==<br />
''To be expanded. <cite>Esser99:ExtensionalityInPositiveTheory</cite>''<br />
<br />
{{References}}<br />
<br />
{{stub}}</div>Wabb2thttp://cantorsattic.info/index.php?title=High-jump&diff=2058High-jump2017-11-11T19:38:34Z<p>Wabb2t: </p>
<hr />
<div>''High-jump'' cardinals are a certain kind of large cardinals. A cardinal $\kappa$ is ''high-jump'' if it is the critical point of an [[elementary embedding]] $j:V\to M$ such that $M$ is closed under sequences of length $sup\{j(f)(\kappa)|f:\kappa\to\kappa\}$. This closure condition is a weakening of the definition of a [[huge]] cardinal.<br />
<br />
== Definition ==<br />
<br />
Let $j:M\to N$ be a (nontrivial) elementary embedding. The ''clearance'' of $j$ is the ordinal $sup\{j(f)(\kappa)|f:\kappa\to\kappa\}$ where $\kappa$ is the critical point of $j$.<br />
<br />
A cardinal $\kappa$ is '''high-jump''' if there exists $j:V\to M$ with critical point $\kappa$ and clearance $\theta$ such that $M^\theta\subseteq M$, i.e. $M$ contains all sequences of elements of $M$ of length $\theta$. $j$ is called a ''high-jump embedding'', and a [[filter|normal fine ultrafilter]] on some $\mathcal{P}_\kappa(\lambda)$ generating an [[ultrapower|ultrapower embedding]] that is high-jump is a ''high-jump ultrafilter'' (or ''high-jump measure'').<br />
<br />
$\kappa$ is called ''almost high-jump'' if $M$ is closed under sequences of length $<\theta$ instead, i.e. $M^\lambda\subseteq M$ for all $\lambda<\theta$. $j$ is then an ''almost high-jump'' embedding. This means that for all $f:\kappa\to\kappa$, $M^{j(f)(\kappa)}\subseteq M$. [[Shelah|Shelah for supercompactness]] cardinals are a natural weakening of almost high-jump cardinals which allows to have one embedding per $f:\kappa\to\kappa$ rather than a single embedding for all such $f$s.<br />
<br />
$\kappa$ is high-jump ''order $\eta$'' (resp. almost high-jump ''order $\eta$'') if there exists a strictly increasing sequence of ordinals $\{\theta_\alpha|\alpha<\eta\}$ such that for all $\alpha<\eta$, there exists a high-jump embedding (resp. almost high-jump embedding) with critical point $\kappa$ and clearance $\theta_\alpha$. $\kappa$ is ''super high-jump'' (resp. ''super almost high-jump'') if there are high-jump embeddings (resp. almost high-jump embeddings) with arbtirarily large clearance (i.e. it is "(almost) high-jump order Ord").<br />
<br />
A high-jump cardinal $\kappa$ has ''unbounded excess closure'' if for some clearance $\theta$, for all cardinals $\lambda\geq\theta$, there is a high-jump measure on $\mathcal{P}_\kappa(\lambda)$ generating an ultrapower embedding with clearance $\theta$.<br />
<br />
The dual notion ''high-jump-for-strongness'', where the closure condition $M^\theta\subseteq M$ is weakened to $V_\theta\subseteq M$, turns out to be equivalent to [[superstrong|superstrongness]].<br />
<br />
== Properties ==<br />
<br />
Let $j:V\to M$ a nontrivial elementary embedding with critical point $\kappa$ and clearance $\theta$. Then there is no $f:\kappa\to\kappa$ such that $j(f)(\kappa)=\theta$.<br />
Also, $\kappa^{+}\leq cf(\theta)\leq 2^\kappa$ (see [[cofinality]]) and $\beth^M_\theta=\theta$. Moreover, $M_\theta\prec M_{j(\kappa)}$ and $M_\theta$ satisfies ZFC where $M_\theta=M\cap V_\theta$.<br />
<br />
When $\kappa$ is almost high-jump, in both $V$ and $M$, $\theta^\kappa=\theta$, also $\theta$ is [[singular]]. Moreover, $V_\theta\prec M_{j(\kappa)}$ and $V_\theta$ satisfies ZFC.<br />
<br />
The following statements also holds:<br />
<br />
* Suppose there is a almost high-jump cardinal. Then there are many cardinals below it that are Shelah for supercompactness. Also, in the model $V_\kappa$ there are many supercompact cardinals.<br />
<br />
* Every high-jump cardinal is almost high-jump, and has order $\theta$; in fact, in the models $V_\theta$, $V_\kappa$ and $M_{j(\kappa)}$ there are many super almost high-jump cardinals.<br />
<br />
* The existence of a high-jump cardinal with order $\eta$ implies that for every $\gamma<\eta$, there exists a model in which that cardinal is high-jump with order $\gamma$. The same statement holds for almost high-jump cardinals.<br />
<br />
* The existence of a high-jump cardinal with unbounded excess closure is equiconsistent with the existence of a cardinal $\kappa$ such that for all sufficiently large $\lambda$, there exists a high-jump measure on $\mathcal{P}_\kappa(\lambda$). <br />
<br />
* Suppose $\kappa$ is [[huge|almost huge]]; then in the model $V_\kappa$ there are many cardinals that are high-jump with unbounded excess closure.<br />
<br />
* Suppose that there exists a pair of cardinals ($\kappa$, $\theta$) such that there is a high-jump embedding $j:V\to M$ with critical point $\kappa$ and clearance $\theta$ and such that $M^{2^\theta}\subseteq M$. Then the cardinal $\kappa$ is super high-jump in the model $V_\theta$, and the cardinal $\kappa$ has high-jump order $\theta$ in $V$. Furthermore, there are many super high-jump cardinals in the models $V_\kappa$, $V_\theta$, and $M_{j(\kappa)}$.<br />
<br />
* The least high-jump cardinal is not $\Sigma_2$-reflecting. In particular, it is not supercompact and not even strong. The same is true for the least [[huge|almost huge]] cardinal, the least [[superstrong]] cardinal, the least almost-high-jump cardinal, and the least Shelah-for-supercompactness cardinal.<br />
<br />
== Name origin ==<br />
<br />
== Read More ==<br />
* Norman Lewis Perlmutter, ''The large cardinals between supercompact and almost-huge'' [http://boolesrings.org/perlmutter/files/2013/07/HighJumpForJournal.pdf]<br />
<br />
{{stub}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Tree_property&diff=2057Tree property2017-11-11T19:37:38Z<p>Wabb2t: </p>
<hr />
<div>An infinite cardinal $\kappa$ has the '''tree property''' if every tree of height $\kappa$ whose levels has cardinality smaller than $\kappa$ has a branch of height $\kappa$ (a cofinal branch). Equivalently, there is no ''$\kappa$-Aronszajn tree'', when a tree is $\kappa$-Aronszajn when it has height $\kappa$, levels with cardinality less than $\kappa$, yet has no cofinal branch.<br />
<br />
== Definition ==<br />
<br />
A ''tree'' is a partially order set (poset) $(T,<)$ such that for all $x\in T$, the order $<$ is a well-order on the set $\{y|y<x\}$ (called a ''chain''). The order type (length) of $<$ on that set is called the ''height'' of $x$. The height of $T$ is the supremum of the heights of all the sets $x\in T$. A ''$\alpha$th level'' of $T$ is a set that contains all $x\in T$ of height $\alpha$. A ''branch'' is a set $B$ well-ordered by $<$ such that any element of $T$ not in $B$ is incomparable with at least one element of $B$.<br />
<br />
A tree is ''$\kappa$-Aronszajn'' if it has height $\kappa$, all its levels have cardinality smaller than $\kappa$, and every branch of $T$ has order type smaller than $\kappa$. An infinite cardinal $\kappa$ has the ''tree property'' if there is no $\kappa$-Aronszajn tree.<br />
<br />
== Properties ==<br />
<br />
''Konig's lemma'' states that $\aleph_0$ has the tree property. It is however provable that $\aleph_1$ does not have the tree property. Cummings and Foreman proved that, under suitable large cardinal assumptions (namely, the existence of many supercompacts), it is consistent with ZFC all $\aleph_n$ cardinals have the tree property for $1<n<\omega$.<br />
<br />
No cardinal can both be a successor cardinal in $L$ and have the tree property in $L$ (the [[constructible universe]]), thus the [[:wikipedia:axiom of constructibility|axiom of constructibility]] is incompatible with the existence of any successor cardinal with the tree property.<br />
<br />
[[Weakly compact]] cardinals all have the tree property. Every cardinal that is [[inaccessible]] and has the tree property is weakly compact. Moreover, every uncountable cardinal with the tree property is weakly compact in the constructible universe, even if it is not inaccessible (in the universe of sets).<br />
<br />
Foreman, Magidor and Schindler showed that if there exists infinitely many cardinals $\delta$ above the continuum such that both $\delta$ and $\delta^{+}$ has the tree property, then the [[axiom of projective determinacy]] holds. This hypothesis was shown to be consistent relative to the existence of infinitely many [[supercompact]] cardinals by Cummings and Foreman.<br />
<br />
Magidor and Shelah showed, from the existence of a [[huge]] cardinals with infinitely many supercompact cardinals above it, the consistency of $\aleph_{\omega+1}$ having the tree property, and furthermore that the successor of a singular limit of [[strongly compact]] cardinals has the tree property.<br />
<br />
== Definable tree property ==<br />
<br />
== Strengthenings of the tree property ==<br />
<br />
== Special Aronszajn trees ==<br />
<br />
{{stub}}</div>Wabb2thttp://cantorsattic.info/index.php?title=HOD&diff=2056HOD2017-11-11T19:37:07Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: HOD}}<br />
HOD denotes the class of ''Hereditarily Ordinal Definable'' sets. It is a definable canonical inner model of ZFC. <br />
<br />
<br />
Although it is definable, this definition is not absolute for transitive inner models of ZF, i.e. given two models $M$ and $N$ of $ZF$, $HOD$ as computed in $M$ may differ from $HOD$ as computed in $N$. <br />
<br />
<br />
==Ordinal Definable Sets==<br />
<br />
Elements of $OD$ are all definable from a finite collection of ordinals. <br />
<br />
==Relativizations==<br />
<br />
<br />
{{References}}<br />
<br />
{{stub}}</div>Wabb2thttp://cantorsattic.info/index.php?title=L_of_V_lambda%2B1&diff=2055L of V lambda+12017-11-11T19:36:42Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: $\exists j:L(V_{\lambda+1})\to L(V_{\lambda+1})$}}<br />
The large cardinal axiom of the title asserts that some non-trivial elementary embedding $j:V_{\lambda+1}\to V_{\lambda+1}$ extends to a non-trivial elementary embedding $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$, where $L(V_{\lambda+1})$ is the transitive proper class obtained by starting with $V_{\lambda+1}$ and forming the constructible hierarchy over $V_{\lambda+1}$ in the usual fashion. An alternate, but equivalent formulation asserts the existence of some non-trivial elementary embedding $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$ with $cr(j) < \lambda$. The critical point assumption is essential for the large cardinal strength as otherwise the axiom would follow from the existence of some measurable cardinal above $\lambda$. The axiom is of [[rank into rank]] type, despite its formulation as an embedding between proper classes, and embeddings witnessing the axiom known as I0 embeddings.<br />
<br />
Originally formulated by Woodin in order to establish the relative consistency of a strong [[determinacy]] hypothesis, it is now known to be obsolete for this purpose (it is far stronger than necessary). Nevertheless, research on the axiom and its variants is still widely pursued and there are numerous intriguing open questions regarding the axiom and its variants, see . <br />
<br />
The axiom subsumes a hierarchy of the strongest large cardinals not known to be inconsistent with ZFC and so is seen as ``straining the limits of consistency" <cite>Kanamori2009:HigherInfinite</cite>. An immediate observation due to the [[Kunen inconsistency]] is that, under the I0 axiom, $L(V_{\lambda+1})$ ''cannot'' satisfy the axiom of choice. <br />
<br />
==The $L(V_{\lambda+1})$ Hierarchy==<br />
<br />
==Relation to the I1 Axiom==<br />
<br />
==Ultrapower Reformulation==<br />
<br />
==Similarities with $L(\mathbb{R})$ under Determinacy==<br />
<br />
<br />
==Strengthenings of I0 and Woodin's $E_\alpha(V_{\lambda+1})$ Sequence==<br />
<br />
{{References}}<br />
<br />
{{stub}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Supercompact&diff=2054Supercompact2017-11-11T19:36:01Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: Supercompact cardinal}}<br />
Supercompact cardinals are best motivated as a generalization of [[measurable]] cardinals, particularly the characterization of measurable cardinals in terms of [[elementary embedding|elementary embeddings]] and strong closure properties. The notion of supercompactness and its consequences was initially developed by Solovay and Reinhardt and further elaborated on by Magidor and Gitik, among many others. Assuming the existence of a supercompact is a very strong assumption and the large cardinal strength of supercompact cardinals is seen in a wide (and bewildering) array of set-theoretic contexts, especially the development of strong forcing axioms and establishing regularity properties of sets of reals. The inner model program has yet to reach the level of a supercompact cardinal and this is considered a prominent open problem in the program itself. Curiously, by results of Woodin, should the inner program reach the level of a supercompact, there is a sense in which it will have reached all greater large cardinals, a startling contrast to previous advances in the program. <br />
<br />
<br />
==Formal definition and equivalent characterizations==<br />
<br />
Generalizing the [[elementary embedding]] characterization of measurable cardinal, a cardinal $\kappa$ is ''$\theta$-supercompact'' if there is an elementary embedding $j:V\to M$ with $M$ a transitive class, such that $j$ has critical point $\kappa$ and $M^\theta\subset M$, i.e. $M$ is closed under arbitrary sequences of length $\theta$. Under the [[axiom of choice]], one may assume without loss of generality that $j(\kappa)\gt\theta$. $\kappa$ is then said to be ''supercompact'' if it is $\theta$-supercompact for all $\theta$. It is worth noting that, using this formulation, $H_{\theta^+}$ must be contained in the transitive class $M$. <br />
<br />
There is an alternative formulation that is expressible in ZFC using certain [[ultrafilter]]s with somewhat technical properties: for $\theta\geq\kappa$, $\kappa$ if $\theta$-supercompact if there is a normal fine measure on $\mathcal{P}_\kappa(\theta)$. $\kappa$ is supercompact if for every set $A$ with $|A|\geq\kappa$, there is a normal fine measure on $\mathcal{P}_\kappa(A)$.<br />
<br />
One can see the equivalence of the two formulations by first considering the ultrafilter $U$ arising from the [[seed]] $j''\theta$, so that $X\in U\iff j''\theta\in j(X)$. It is easy to check that $U$ is a normal fine measure on $\mathcal{P}_\kappa(\theta)$. Conversely, the ultrapower by a normal fine measure $U$ on $\mathcal{P}_\kappa(\theta)$ gives rise to an embedding $j:V\to M$ (here $M$ is identified with the transitive collapse of the ultrapower by $U$). It is then straightforward to check that $\theta$ is the critical point of this embedding and that $M$ is sufficiently closed, thus witnessing $\theta$-supercompactness of $\kappa$. <br />
<br />
A third characterization was given by Magidor in terms of elementary embeddings from initial segments of $V$ into other (larger) initial segments of $V$, but in this characterization, the supercompact cardinal $\kappa$ is the ''image'' of the critical point of this embedding, rather than the critical point itself: $\kappa$ is supercompact if and only if for every $\eta>\kappa$ there is $\alpha<\kappa$ such that there exists a nontrivial elementary embedding $j:V_\alpha\to V_\eta$ such that $j(crit(j))=\kappa$.<br />
<br />
== Properties ==<br />
<br />
If $\kappa$ is supercompact, then there are $2^{2^\kappa}$ [[filter|normal fine measures]] on $\kappa$, also for every $\lambda\geq\kappa$ there are $2^{2^{\lambda^{<\kappa}}}$ normal fine measures on $\mathcal{P}_\kappa(\lambda)$.<br />
<br />
Every supercompact has [[Mitchell order]] $(2^\kappa)^+\geq\kappa^{++}$.<br />
<br />
If $\kappa$ is $\lambda$-supercompact then it is also $\mu$-supercompact for every $\mu<\lambda$. If $\lambda\geq\kappa$ is regular, $\kappa$ is $\lambda$-supercompact, then every $\alpha<\kappa$ that is $\gamma$-supercompact for all $\gamma<\kappa$ (if any exists) is also $\lambda$-supercompact. In the same vein, for every cardinals $\kappa<\lambda$, if $\lambda$ is supercompact and $\kappa$ is $\gamma$-supercompact for all $\gamma<\lambda$, then $\kappa$ is also supercompact.<br />
<br />
''Laver's theorem'' asserts that if $\kappa$ is supercompact, there exists a function $f:\kappa\to V_\kappa$ such that for every $x$ and $\lambda\geq\kappa$ with $|tc(x)|\leq\lambda$ there exists a normal fine measure $U$ on $\mathcal{P}_\kappa(\lambda)$ such that $j_U(f)(\kappa)=x$, where $j_U$ is the elementary embedding generated from $U$. Here $tc(x)$ is the ''transitive closure'' of $x$ (i.e. the smallest transitive set containing $x$), and $f$ is called a ''Laver function''.<br />
<br />
===Reflection Properties===<br />
<br />
== Supercompact cardinals and forcing ==<br />
<br />
=== The continuum hypothesis and supercompact cardinals ===<br />
<br />
If $\kappa$ is $\lambda$-supercompact and $2^\alpha=\alpha^{+}$ for every $\alpha<\kappa$, then $2^\alpha=\alpha^{+}$ for every $\alpha\leq\lambda$. Consequently, if the [[GCH|generalized continuum hypothesis]] holds below a supercompact cardinal, then it holds everywhere.<br />
<br />
The existence of a supercompact implies the consistency of the failure of the ''singular cardinal hypothesis'', i.e. it is consistent that the generalized continuum hypothesis fails at a strong limit singular cardinal. It also implies the consistency of the failure of the GCH at a measurable cardinal.<br />
<br />
By combining results of Magidor, Shelah and Gitik, one can show that the existence of a supercompact also implies the existence of a generic extension in which $2^{\aleph_\alpha}<\aleph_{\omega_1}$ for all $\alpha<\omega_1$, but also $2^{\aleph_{\omega_1}}>\aleph_{\omega_1+\alpha+1}$ for any prescribed $\alpha<\omega_2$. Similarly, one can have a generic extension in which the GCH holds below $\aleph_\omega$ but $2^{\aleph_\omega}>\aleph_{\omega+\alpha+1}$ for any prescribed $\alpha<\omega_1$.<br />
<br />
Woodin and Cummings furthermore showed that if there exists a supercompact, then there is a generic extension in which $2^\kappa=\kappa^{++}$ for ''every'' cardinal $\kappa$, i.e. the GCH fails ''everywhere''(!).<br />
<br />
=== Laver preparation ===<br />
<br />
''Indestructibility, including the Laver diamond.''<br />
<br />
=== Proper forcing axiom ===<br />
<br />
Baumgartner proved that if there is a supercompact cardinal, then the proper forcing axiom holds in a forcing extenion. PFA's strengthening, PFA$^{+}$, is also consistent relative to the existence of a supercompact cardinal.<br />
<br />
=== Martin's Maximum ===<br />
<br />
== Relation to other large cardinals ==<br />
<br />
Every cardinal $\kappa$ that is $2^\kappa$-supercompact is the $\kappa$th [[superstrong]] cardinal, and is preceeded by a stationary set of superstrongs.<br />
<br />
If a cardinal $\theta$-supercompact then it also $\theta$-[[strongly compact]]. Consequently, every supercompact cardinal is also strongly compact. It is consistent with $ZFC$ that every strongly compact cardinal is also supercompact, but it is not currently known whether the existence of a strongly compact cardinal is equiconsistent with the existence of a supercompact cardinal. If $\kappa$ is supercompact, then there is a forcing extension in which $\kappa$ remains supercompact and is also the least strongly compact cardinal. If there exists a measurable cardinal that is a limit of strongly compact cardinals, then the least such cardinal is strongly compact but not supercompact, in fact not even $2^\kappa$-supercompact.<br />
<br />
Under the [[axiom of determinacy]], $\omega_1$ is <$\Theta$-supercompact, where $\Theta$ is at least an [[aleph fixed point]], and under $V=L(\mathbb{R})$ is even weakly $\Theta$-[[Mahlo]]. The existence of a supercompact cardinals also implies the axiom ${L(\mathbb{R})}$-determinacy.<br />
<br />
If $\kappa$ is $|V_{\kappa+\eta}|$-supercompact with $\eta<\kappa$ then it is preceeded by a stationary set of $\eta$-[[extendible]] cardinals. If $\kappa$ is $(\eta+2)$-extendible then it is $|V_{\kappa+\eta}|$-supercompact. The least supercompact is not 1-extendible, in fact any cardinal that is both supercompact and 1-extendible is preceeded by a stationary set of cardinals that are both supercompact and limits of supercompact cardinals.<br />
<br />
The least supercompact is larger than the least [[huge]] cardinal (if such a cardinal exists). It is also larger than the least n-huge cardinal, for all n. If $\kappa$ is supercompact and there is an n-huge cardinal above $\kappa$, then there are $\kappa$-many n-huge cardinals below $\kappa$.<br />
<br />
{{stub}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Zero_sharp&diff=2053Zero sharp2017-11-11T19:34:30Z<p>Wabb2t: </p>
<hr />
<div>[[Category:Large cardinal axioms]]<br />
[[Category:Constructibility]]<br />
$0^{\#}$ is a [[projective|$\Sigma_3^1$]] real number which cannot be proven to exist in [[ZFC]]. It's existence contradicts the [[Axiom of constructibility]], $V=L$. In fact, it's existence is somewhat equivalent to $L$ being completely different from $V$. <br />
<br />
== Definition ==<br />
<br />
$0^{\#}$ is defined as the set of all Gödel numberings of first-order formula $\varphi$ such that $L\models\varphi(\aleph_0,\aleph_1...\aleph_n)$ for some $n$. Because of the [[stable|stability]] of $\aleph_\omega$, $0^{\#}$ is equivalent to the set of all Gödel numberings of first-order formula $\varphi$ such that $L_{\aleph_{\omega}}\models\varphi(\aleph_0,\aleph_1...\aleph_n)$. This definition implies the existence of Silver Indiscernables. Moreover, it implies:<br />
<br />
*Given any set $X\in L$ which is first-order definable in $L$, $X\in L_{\omega_1}$. This of course implies that $\aleph_1$ is not first-order definable in $L$, because $\aleph_1\not\in L_{\omega_1}$. This is already a disproof of $V=L$ (because $\aleph_1$ is first-order definable).<br />
*For every $\alpha\in\omega_1^L$, every uncountable cardinal is [[Ramsey#iterable|$\alpha$-iterable]], $\geq$ an [[Erdos|$\alpha$-Erdős]], and [[ineffable|totally ineffable]] in $L$.<br />
*There are $\mathfrak{c}$ many reals which are not constructible (that is, $x\not\in L$).<br />
<br />
The existence of $0^\#$ is implied by:<br />
* Chang's Conjecture.<br />
* The existence of an $\omega_1$-iterable cardinal.<br />
* The negation of the singular cardinal hypothesis (SCH).<br />
* The [[axiom of determinacy]] (AD).<br />
<br />
== $0^{\#}$ cardinal ==<br />
<br />
$0^{\#}$ exists iff there is a nontrivial [[elementary embedding]] $j:L\rightarrow L$ (by a theorem of Kunen). The critical point of such an embedding is sometimes called a $0^{\#}$ cardinal, and sometimes called a $j:L\rightarrow L$ cardinal. These cardinals do not coincide with measurable cardinals for a long time. While the least measurable cardinal is [[indescribable|$\Sigma_1^2$-describable]], each of these cardinals is totally indescribable. Furthermore, the least measurable cardinal $\kappa$ such that $V_\kappa$ satisfies the existence of a measurable cardinal is not a $j:L\rightarrow L$ cardinal, and the least measurable cardinal $\kappa$ such that $V_\kappa$ satisfies the existence of such a cardinal is not a $j:L\rightarrow L$ cardinal, and so on.<br />
<br />
However, the existence of a measurable suffices to prove the existence and consistency of a $j:L\rightarrow L$ cardinal.<br />
<br />
''More information to be added here.''<br />
<br />
== References ==<br />
*Jech, Thomas J. Set Theory (The 3rd Millennium Ed.). Springer, 2003.</div>Wabb2thttp://cantorsattic.info/index.php?title=Rank_into_rank&diff=2052Rank into rank2017-11-11T19:33:52Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: Rank into rank}}<br />
A nontrivial [[elementary embedding]] $j:V_\lambda\to V_\lambda$ for some infinite ordinal $\lambda$ is known as a ''rank into rank embedding'' and the axiom asserting that such an embedding exists is usually denoted by I3, I2, I1, $\mathcal{E}(V_\lambda)\neq \emptyset$ or some variant thereof. The term applies to a hierarchy of such embeddings increasing in large cardinal strength reaching toward the [[Kunen inconsistency]]. The axioms in this section are in some sense a technical restriction falling out of Kunen's proof that there can be no nontrivial elementary embedding $j:V\to V$ in ZFC). An analysis of the proof shows that there can be no nontrivial $j:V_{\lambda+2}\to V_{\lambda+2}$ and that if there is some ordinal $\delta$ and nontrivial rank to rank embedding $j:V_\delta\to V_\delta$ then necessarily $\delta$ must be a strong limit cardinal of cofinality $\omega$ or the successor of one. By standing convention, it is assumed that rank into rank embeddings are not the identity on their domains.<br />
<br />
There are really two cardinals relevant to such embeddings: The large cardinal is the critical point of $j$, often denoted $crit(j)$ or sometimes $\kappa_0$, and the other (not quite so large) cardinal is $\lambda$. In order to emphasize the two cardinals, the axiom is sometimes written as E($\kappa$,$\lambda$) (or I3($\kappa$,$\lambda$), etc.) as in <cite>Kanamori2009:HigherInfinite</cite>. The cardinal $\lambda$ is determined by defining the ''critical sequence of $j$''. Set $\kappa_0 = crit(j)$ and $\kappa_{n+1}=j(\kappa_n)$. Then $\lambda = \sup \langle \kappa_n : n <\omega\rangle$ and is the first fixed point of $j$ that occurs above $\kappa_0$. Note that, unlike many of the other large cardinals appearing in the literature, the ordinal $\lambda$ is ''not'' the target of the critical point; it is the $\omega^{th}$ $j$-iterate of the critical point. <br />
<br />
As a result of the strong closure properties of rank into rank embeddings, their critical points are [[huge]] and in fact $n$-huge for every $n$. This aspect of the large cardinal property is often called $\omega$-hugeness and the term ''$\omega$-huge cardinal'' is sometimes used to refer to the critical point of some rank into rank embedding. <br />
<br />
==The I3 Axiom and Natural Strengthenings== <br />
<br />
The I3 axiom asserts, generally, that there is some embedding $j:V_\lambda\to V_\lambda$. I3 is also denoted as $\mathcal{E}(V_\lambda)\neq\emptyset$ where $\mathcal{E}(V_\lambda)$ is the set of all elementary embeddings from $V_\lambda$ to $V_\lambda$, or sometimes even I3($\kappa$,$\lambda$) when mention of the relevant cardinals is necessary. In its general form, the axiom asserts that the embedding preserves all first-order structure but fails to specify how much second-order structure is preserved by the embedding. The case that ''no'' second-order structure is preserved is also sometimes denoted by I3. In this specific case I3 denotes the weakest kind of rank into rank embedding and so the I3 notation for the axiom is somewhat ambiguous. To eliminate this ambiguity we say $j$ is $E_0(\lambda)$ when $j$ preserves only first-order structure. <br />
<br />
The axiom can be strengthened and refined in a natural way by asserting that various degrees of second-order correctness are preserved by the embeddings. A rank into rank embedding $j$ is said to be $\Sigma^1_n$ or ''$\Sigma^1_n$ correct'' if, for every $\Sigma^1_n$ formula $\Phi$ and $A\subseteq V_\lambda$ the elementary schema holds for $j,\Phi$, and $A$: $$V_\lambda\models\Phi(A) \Leftrightarrow V_\lambda\models\Phi(j(A)).$$<br />
The more specific axiom $E_n(\lambda)$ asserts that some $j\in\mathcal{E}(V_\lambda)$ is $\Sigma^1_{2n}$. <br />
<br />
The ``$2n$" subscript in the axiom $E_n(\lambda)$ is incorporated so that the axioms $E_m(\lambda)$ and $E_n(\lambda)$ where $m<n$ are strictly increasing in strength. This is somewhat subtle. For $n$ odd, $j$ is $\Sigma^1_n$ if and only if $j$ is $\Sigma^1_{n+1}$. However, for $n$ even, $j$ being $\Sigma^1_{n+1}$ is ''significantly'' stronger than a $j$ being $\Sigma^1_n$<cite>Laver1997:Implications</cite>. <br />
<br />
==The I2 Axiom==<br />
<br />
Any $j:V_\lambda\to V_\lambda$ can be extended to a $j^+:V_{\lambda+1}\to V_{\lambda+1}$ but in only one way: Define for each $A\subseteq V_\lambda$ $$j^+(A)=\bigcup_{\alpha < \lambda}(j(V_\alpha\cap A)).$$ $j^+$ is not necessarily elementary. The I2 axiom asserts the existence of some elementary embedding $j:V\to M$ with $V_\lambda\subseteq M$ where $\lambda$ is defined as the $\omega^{th}$ $j$-iterate of the critical point. Although this axiom asserts the existence of a ''class'' embedding with a very strong closure property, it is in fact equivalent to an embedding $j:V_\lambda\to V_\lambda$ with $j^+$ preserving well-founded relations on $V_\lambda$. So this axioms preserves ''some'' second-order structure of $V_\lambda$ and is in fact equivalent to $E_1(\lambda)$ in the hierarchy defined above. A specific property of I2 embeddings is that they are ''iterable'' (i.e. the direct limit of directed system of embeddings is well-founded). In the literature, IE($\lambda$) asserts that $j:V_\lambda\to V_\lambda$ is iterable and IE($\lambda$) falls strictly between $E_0(\lambda)$ and $E_1(\lambda)$. <br />
<br />
As a result of the strong closure property of I2, the equivalence mentioned above cannot be through an analysis of some ultrapower embedding. Instead, the equivalence is established by constructing a directed system of embeddings of various ultrapowers and using reflection properties of the critical points of the embeddings. The direct limit is well-founded since well-founded relations are preserved by $j^+$. The use of both direct and indirect limits, in conjunction with reflection arguments, is typical for establishing the properties of rank into rank embeddings. <br />
<br />
==The I1 Axiom==<br />
<br />
I1 asserts the existence of a nontrivial elementary embedding $j:V_{\lambda+1}\to V_{\lambda+1}$. This axiom is sometimes denoted $\mathcal{E}(V_{\lambda+1})\neq\emptyset$. Any such embedding preserves all second-order properties of $V_\lambda$ and so is $\Sigma^1_n$ for all $n$. To emphasize the preservation of second-order properties, the axiom is also sometimes written as $E_\omega(\lambda)$. In this case, restricting the embedding to $V_\lambda$ and forming $j^+$ as above yields the original embedding. <br />
<br />
Strengthening this axiom in a natural way leads the I0 axiom, i.e. asserting that embeddings of the form [[L of V lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]] exist. There are also other natural strengthenings of I1, though it is not entirely clear how they relate to the I0 axiom. For example, one can assume the existence of elementary embeddings satisfying I1 which extend to embeddings $j:M\to M$ where $M$ is a transitive class inner model and add various requirements to $M$. These requirements must not entail that $M$ satisfies the axiom of choice by the Kunen inconsistency. Requirements that have been considered include assuming $M$ contains $V_{\lambda+1}$, $M$ satisfies DC$_\lambda$, $M$ satisfies replacement for formulas containing $j$ as a parameter, $j(crit(j))$ is arbitrarily large in $M$, etc. <br />
<br />
==Large Cardinal Properties of Critical Points==<br />
<br />
The critical points of rank into rank embeddings have many strong reflection properties. They are measurable, $n$-huge for all $n$ (hence the terminology $\omega$-huge mentioned in the introduction) and partially supercompact. <br />
<br />
Using $\kappa_0$ as a seed, one can form the ultrafilter $$U=\{X\subseteq\mathcal{P}(\kappa_0): j"\kappa_0\in j(X)\}.$$ Thus, $\kappa_0$ is a measurable cardinal.<br />
<br />
In fact, for any $n$, $\kappa_0$ is also $n$-huge as witnessed by the ultrafilter <br />
$$U=\{X\subseteq\mathcal{P}(\kappa_n): j"\kappa_n\in j(X)\}.$$ This motivates the term $\omega$-huge cardinal mentioned in the introduction. <br />
<br />
Letting $j^n$ denote the $n^{th}$ iteration of $j$, then <br />
$$V_\lambda\models ``\lambda\text{ is supercompact"}.$$ To see this, suppose $\kappa_0\leq \theta <\kappa_n$, then $$U=\{X\subseteq\mathcal{P}_{\kappa_0}(\theta): j^n"\theta\in j^n(X)\}$$ winesses the $\theta$-compactness of $\kappa_0$ (in $V_\lambda$). For this last claim, it is enough that $\kappa_0(j)$ is $<\lambda$-supercompact, i.e. not *fully* supercompact in $V$. In this case, however, $\kappa_0$ *could* be fully supercompact. <br />
<br />
Critical points of rank-into-rank embeddings also exhibit some *upward* reflection properties. For example, if $\kappa$ is a critical point of some embedding witnessing I3($\kappa$,$\lambda$), then there must exist another embedding witnessing I3($\kappa'$,$\lambda$) with critical point ''above'' $\kappa$! This upward type of reflection is not exhibited by large cardinals below [[extendible]] cardinals in the large cardinal hierarchy.<br />
<br />
==Algebras of elementary embeddings==<br />
<br />
If $j,k\in\mathcal{E}_{\lambda}$, then $j^+(k)\in\mathcal{E}_{\lambda}$ as well. We therefore define a binary operation $*$ on $\mathcal{E}_{\lambda}$ called application defined by $j*k=j^{+}(k)$. The binary operation $*$ together with composition $\circ$ satisfies the following identities:<br />
<br />
1. $(j\circ k)\circ l=j\circ(k\circ l),\,j\circ k=(j*k)\circ j,\,j*(k*l)=(j\circ k)*l,\,j*(k\circ l)=(j*k)\circ(j*l)$<br />
<br />
2. $j*(k*l)=(j*k)*(j*l)$ (self-distributivity).<br />
<br />
Identity 2 is an algebraic consequence of the identities in 1.<br />
<br />
If $j\in\mathcal{E}_{\lambda}$ is a nontrivial elementary embedding, then $j$ freely generates a subalgebra of $(\mathcal{E}_{\lambda},*,\circ)$ with respect to the identities in 1, and $j$ freely generates a subalgebra of $(\mathcal{E}_{\lambda},*)$ with respect to the identity 2.<br />
<br />
If $j_{n}\in\mathcal{E}_{\lambda}$ for all $n\in\omega$, then $\sup\{\textrm{crit}(j_{0}*\dots*j_{n})\mid n\in\omega\}=\lambda$ where the implied parentheses a grouped on the left (for example, $j*k*l=(j*k)*l$).<br />
<br />
Suppose now that $\gamma$ is a limit ordinal with $\gamma<\lambda$. Then define an equivalence relation $\equiv^{\gamma}$ on $\mathcal{E}_{\lambda}$ where $j\equiv^{\gamma}k$ if and only if $j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$ for each $x\in V_{\gamma}$. Then the equivalence relation $\equiv^{\gamma}$ is a congruence on the algebra $(\mathcal{E}_{\lambda},*,\circ)$. In other words, if $j_{1},j_{2},k\in \mathcal{E}_{\lambda}$ and $j_{1}\equiv^{\gamma}j_{2}$ then $j_{1}\circ k\equiv^{\gamma} j_{2}\circ k$ and $j_{1}*k\equiv^{\gamma}j_{2}*k$, and if $j,k_{1},k_{2}\in\mathcal{E}_{\lambda}$ and $k_{1}\equiv^{\gamma}k_{2}$ then $j\circ k_{1}\equiv^{\gamma}j\circ k_{2}$ and $j*k_{1}\equiv^{j(\gamma)}j*k_{2}$.<br />
<br />
If $\gamma<\lambda$, then every finitely generated subalgebra of $(\mathcal{E}_{\lambda}/\equiv^{\gamma},*,\circ)$ is finite.<br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Huge&diff=2051Huge2017-11-11T19:07:14Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: Huge cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
[[Category:Critical points]]<br />
Huge cardinals (and their variants) were introduced by Kenneth Kunen in 1972 as a very large cardinal axiom. Kenneth Kunen first used them to prove that the consistency of the existence of a huge cardinal implies the consistency of ZFC+"there is a $\aleph_2$-[[filter|saturated ideal]] over $\omega_1$". <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
== Definitions ==<br />
<br />
Their formulation is similar to that of the formulation of [[superstrong]] cardinals. A huge cardinal is to a [[supercompact]] cardinal as a superstrong cardinal is to a [[strong]] cardinal, more precisely. The definition is part of a generalized phenomenon known as the "double helix", in which for some large cardinal properties $n$-$P_0$ and $n$-$P_1$, $n$-$P_0$ has less consistency strength than $n$-$P_1$, which has less consistency strength than $(n+1)$-$P_0$, and so on. This phenomenon is seen only around the [[n-fold variants|$n$-fold variants]] as of modern set theoretic concerns. <cite>Kentaro2007:DoubleHelix</cite><br />
<br />
Although they are very large, there is a first-order definition which is equivalent to $n$-hugeness, so the $\theta$-th $n$-huge cardinal is first-order definable whenever $\theta$ is first-order definable. This definition can be seen as a (very strong) strengthening of the first-order definition of [[measurable|measurability]].<br />
<br />
=== Elementary embedding definitions ===<br />
<br />
The elementary embedding definitions are somewhat standard. Let $j:V\rightarrow M$ be an [[elementary embedding]] with critical point $\kappa$ such that $M$ is a standard inner model of [[ZFC]]. Then:<br />
<br />
*$\kappa$ is '''almost $n$-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length less than $\lambda$ (that is, $M^{<\lambda}\subset M$).<br />
*$\kappa$ is '''$n$-huge with target $\lambda$''' iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length $\lambda$ ($M^\lambda\subset M$).<br />
*$\kappa$ is '''almost $n$-huge''' iff it is almost $n$-huge with target $\lambda$ for some $\lambda$.<br />
*$\kappa$ is '''$n$-huge''' iff it is $n$-huge with target $\lambda$ for some $\lambda$.<br />
*$\kappa$ is '''super almost $n$-huge''' iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is almost $n$-huge with target $\lambda$ (that is, the target can be made arbitrarily large).<br />
*$\kappa$ is '''super $n$-huge''' iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is $n$-huge with target $\lambda$.<br />
*$\kappa$ is '''almost huge''', '''huge''', '''super almost huge''', and '''superhuge''' iff it is '''almost $1$-huge''', '''$1$-huge''', etc. respectively.<br />
<br />
=== Ultrafilter definition ===<br />
<br />
The first-order definition of $n$-huge is somewhat similar to [[measurable|measurability]]. Specifically, $\kappa$ is measurable iff there is a nonprincipal $\kappa$-complete [[filter|ultrafilter]], $U$, over $\kappa$. A cardinal $\kappa$ is $n$-huge iff there is some cardinal $\lambda$, a nonprincipal $\kappa$-complete ultrafilter, $U$, over $\mathcal{P}(\lambda)$, and cardinals $\kappa=\lambda_0<\lambda_1<\lambda_2...<\lambda_{n-1}<\lambda_n=\lambda$ such that:<br />
<br />
$$\forall i<n\forall x\subseteq\lambda(ot(x\cap\lambda_{i+1})=\lambda_i\rightarrow x\in U)$$<br />
<br />
Where $ot(X)$ is the [[Order-isomorphism|order-type]] of the poset $(X,\in)$. <cite>Kanamori2009:HigherInfinite</cite> This definition is, more intuitively, making $U$ very large, like most ultrafilter characterizations of large cardinals ([[supercompact]], [[strongly compact]], etc.).<br />
<br />
== Consistency strength and size ==<br />
<br />
Hugeness exhibits a phenomenon associated with similarly defined large cardinals (the [[n-fold variants|$n$-fold variants]]) known as the ''double helix''. This phenomenon is when for one $n$-fold variant, letting a cardinal be called $n$-$P_0$ iff it has the property, and another variant, $n$-$P_1$, $n$-$P_0$ is weaker than $n$-$P_1$, which is weaker than $(n+1)$-$P_0$. <cite>Kentaro2007:DoubleHelix</cite> In the consistency strength hierarchy, here is where these lay (top being weakest):<br />
<br />
* [[measurable]] = $0$-[[superstrong]] = almost $0$-huge = super almost $0$-huge = $0$-huge = super $0$-huge <br />
* $n$-superstrong<br />
* $n$-fold supercompact<br />
* $(n+1)$-fold strong, $n$-fold extendible<br />
* $(n+1)$-fold Woodin, $n$-fold Vopěnka<br />
* $(n+1)$-fold Shelah<br />
* almost $n$-huge<br />
* super almost $n$-huge<br />
* $n$-huge<br />
* super $n$-huge<br />
* $(n+1)$-superstrong<br />
<br />
All huge variants lay at the top of the double helix restricted to some [[Omega|natural number]] $n$, although each are bested by [[rank-into-rank|I3]] cardinals (the [[elementary embedding|critical points]] of the I3 elementary embeddings). In fact, every I3 is preceeded by a stationary set of $n$-huge cardinals, for all $n$. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
Similarly, every huge cardinal $\kappa$ is almost huge, and there is a normal measure over $\kappa$ which contains every almost huge cardinal $\lambda<\kappa$. Every superhuge cardinal $\kappa$ is [[extendible]] and there is a normal measure over $\kappa$ which contains every extendible cardinal $\lambda<\kappa$. Every $(n+1)$-huge cardinal $\kappa$ has a normal measure which contains every cardinal $\lambda$ such that $V_\kappa\models$"$\lambda$ is super $n$-huge". <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
In terms of size, however, the least $n$-huge cardinal is smaller than the least [[supercompact]] cardinal. Assuming both exist, for any $\kappa$ which is supercompact and has an $n$-huge cardinal above it, there are $\kappa$ many $n$-huge cardinals less than $\kappa$. <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
Every $n$-huge cardinal is $m$-huge for every $m\leq n$. Similarly with almost $n$-hugeness, super $n$-hugeness, and super almost $n$-hugeness. Every almost huge cardinal is [[Vopenka|Vopěnka]] (therefore the consistency of the existence of an almost-huge cardinal implies the consistency of Vopěnka's principle). <cite>Kanamori2009:HigherInfinite</cite><br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Wholeness_axioms&diff=2050Wholeness axioms2017-11-11T19:06:31Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: The Wholeness Axioms}}<br />
<br />
The wholeness axioms, proposed by Paul Corazza <cite>Corazza2000:WholenessAxiomAndLaverSequences, Corazza2003:GapBetweenI3andWA</cite>, occupy a<br />
high place in the upper stratosphere of the large cardinal<br />
hierarchy, intended as slight weakenings of the [[Kunen inconsistency]], but similar in spirit. <br />
<br />
The ''wholeness axioms'' are formalized in the<br />
language $\{\in,j\}$, augmenting the usual language of set<br />
theory $\{\in\}$ with an additional unary function symbol $j$<br />
to represent the [[elementary embedding]]. The base theory ZFC is<br />
expressed only in the smaller language $\{\in\}$. Corazza's<br />
original proposal, which we denote by WA$_0$, asserts<br />
that $j$ is a nontrivial amenable elementary embedding<br />
from the universe to itself. Elementarity is expressed by<br />
the scheme $\varphi(x)\iff\varphi(j(x))$, where $\varphi$<br />
runs through the formulas of the usual language of set<br />
theory; nontriviality is expressed by the sentence $\exists<br />
x j(x)\not=x$; and amenability is simply the assertion<br />
that $j\upharpoonright A$ is a set for every set $A$. Amenability in this case is equivalent to<br />
the assertion that the separation axiom holds for<br />
$\Delta_0$ formulae in the language $\{\in,j\}$. <br />
The wholeness axiom WA, also denoted WA$_\infty$, asserts in addition that the<br />
full separation axiom holds in the language $\{\in,j\}$. <br />
<br />
Those two axioms are the endpoints of the hierarchy of axioms WA$_n$, asserting increasing amounts of the separation axiom. <br />
Specifically, the wholeness axiom WA$_n$, where $n$ is<br />
amongst $0,1,\ldots,\infty$, consists of the following:<br />
<br />
# (elementarity) All instances of $\varphi(x)\iff\varphi(j(x))$ for $\varphi$ in the language $\{\in,j\}$.<br />
# (separation) All instances of the Separation Axiom for $\Sigma_n$ formulae in the full language $\{\in,j\}$.<br />
# (nontriviality) The axiom $\exists x\,j(x)\not=x$.<br />
<br />
Clearly, this resembles the [[Kunen inconsistency]]. What is missing from the wholeness<br />
axiom schemes, and what figures prominantly in Kunen's<br />
proof, are the instances of the replacement axiom in the<br />
full language with $j$. In particular, it is the replacement axiom in the language with $j$ that allows one to define the critical sequence $\langle \kappa_n\mid n\lt\omega\rangle$, where $\kappa_{n+1}=j(\kappa_n)$, which figures in all the proofs of the Kunen inconsistency. Thus, none of the proofs of the Kunen inconsistency can be carried out with WA, and indeed, in every model of WA the critical sequence is unbounded in the ordinals. <br />
<br />
The hiearchy of wholeness axioms is strictly increasing in strength, if consistent. <cite>Hamkins2001:WholenessAxiomAndVequalHOD</cite> <br />
<br />
If $j:V_\lambda\to V_\lambda$ witnesses a [[rank into rank]] cardinal, then $\langle V_\lambda,\in,j\rangle$ is a model of the wholeness axiom.<br />
<br />
If the wholeness axiom is consistent with ZFC, then it is consistent with ZFC+V=HOD.<cite>Hamkins2001:WholenessAxiomAndVequalHOD</cite><br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Weakly_compact&diff=2049Weakly compact2017-11-11T19:05:44Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: Weakly compact cardinal}}<br />
[[Category:Large cardinal axioms]]<br />
Weakly compact cardinals lie at the focal point of a number<br />
of diverse concepts in infinite combinatorics, admitting various characterizations in terms of these concepts. If $\kappa^{{<}\kappa} = \kappa$, then the following are equivalent: <br />
<br />
:; Weak compactness : A cardinal $\kappa$ is weakly compact if and only if it is [[uncountable]] and every $\kappa$-satisfiable theory in an [[Infinitary logic|$\mathcal{L}_{\kappa,\kappa}$]] language of size at most $\kappa$ is satisfiable.<br />
:; Extension property : A cardinal $\kappa$ is weakly compact if and only if for every $A\subset V_\kappa$, there is a transitive structure $W$ properly extending $V_\kappa$ and $A^*\subset W$ such that $\langle V_\kappa,{\in},A\rangle\prec\langle W,{\in},A^*\rangle$.<br />
:; Tree property : A cardinal $\kappa$ is weakly compact if and only if it is [[inaccessible]] and has the [[tree property]].<br />
:; Filter property : A cardinal $\kappa$ is weakly compact if and only if whenever $M$ is a set containing at most $\kappa$-many subsets of $\kappa$, then there is a $\kappa$-[[filter|complete nonprincipal filter]] $F$ measuring every set in $M$.<br />
:; Weak embedding property : A cardinal $\kappa$ is weakly compact if and only if for every $A\subset\kappa$ there is a transitive set $M$ of size $\kappa$ with $\kappa\in M$ and a transitive set $N$ with an [[elementary embedding|embedding]] $j:M\to N$ with [[critical point]] $\kappa$.<br />
:; Embedding characterization : A cardinal $\kappa$ is weakly compact if and only if for every transitive set $M$ of size $\kappa$ with $\kappa\in M$ there is a transitive set $N$ and an embedding $j:M\to N$ with critical point $\kappa$.<br />
:; Normal embedding characterization : A cardinal $\kappa$ is weakly compact if and only if for every $\kappa$-model $M$ there is a $\kappa$-model $N$ and an embedding $j:M\to N$ with critical point $\kappa$, such that $N=\{\ j(f)(\kappa)\mid f\in M\ \}$.<br />
;; Hauser embedding characterization : A cardinal $\kappa$ is weakly compact if and only if for every $\kappa$-model $M$ there is a $\kappa$-model $N$ and an embedding $j:M\to N$ with critical point $\kappa$ such that $j,M\in N$.<br />
:; Partition property : A cardinal $\kappa$ is weakly compact if and only if it enjoys the partition property $\kappa\to(\kappa)^2_2$.<br />
:; Indescribability property : A cardinal $\kappa$ is weakly compact if and only if it is $\Pi_1^1$-[[indescribable]].<br />
<br />
Weakly compact cardinals first arose<br />
in connection with (and were named for) the question of<br />
whether certain [[Infinitary logic|infinitary logics]] satisfy the compactness<br />
theorem of first order logic. Specifically, in a language<br />
with a signature consisting, as in the first order context,<br />
of a set of constant, finitary function and relation<br />
symbols, we build up the language of $\mathcal{L}_{\kappa,\lambda}$<br />
formulas by closing the collection of formulas under<br />
infinitary conjunctions<br />
$\wedge_{\alpha<\delta}\varphi_\alpha$ and disjunctions<br />
$\vee_{\alpha<\delta}\varphi_\alpha$ of any size<br />
$\delta<\kappa$, as well as infinitary quantification<br />
$\exists\vec x$ and $\forall\vec x$ over blocks of<br />
variables $\vec x=\langle x_\alpha\mid\alpha<\delta\rangle$ of size less<br />
than $\kappa$. A theory in such a language is ''satisfiable'' if it has a model under the natural semantics.<br />
A theory is ''$\theta$-satisfiable'' if every subtheory<br />
consisting of fewer than $\theta$ many sentences of it is<br />
satisfiable. First order logic is precisely<br />
$L_{\omega,\omega}$, and the classical Compactness theorem<br />
asserts that every $\omega$-satisfiable $\mathcal{L}_{\omega,\omega}$<br />
theory is satisfiable. A uncountable cardinal $\kappa$ is<br />
''[[strongly compact]]'' if every $\kappa$-satisfiable<br />
$\mathcal{L}_{\kappa,\kappa}$ theory is satisfiable. The cardinal<br />
$\kappa$ is ''weakly compact'' if every<br />
$\kappa$-satisfiable $\mathcal{L}_{\kappa,\kappa}$ theory, in a<br />
language having at most $\kappa$ many constant, function<br />
and relation symbols, is satisfiable.<br />
<br />
Next, for any cardinal $\kappa$, a ''$\kappa$-tree'' is a<br />
tree of height $\kappa$, all of whose levels have size less<br />
than $\kappa$. More specifically, $T$ is a ''tree'' if<br />
$T$ is a partial order such that the predecessors of any<br />
node in $T$ are well ordered. The $\alpha^{\rm th}$ level of a<br />
tree $T$, denoted $T_\alpha$, consists of the nodes whose<br />
predecessors have order type exactly $\alpha$, and these<br />
nodes are also said to have ''height'' $\alpha$. The height of the tree $T$ is the first $\alpha$ for which $T$<br />
has no nodes of height $\alpha$. A ""$\kappa$-branch""<br />
through a tree $T$ is a maximal linearly ordered subset of<br />
$T$ of order type $\kappa$. Such a branch selects exactly<br />
one node from each level, in a linearly ordered manner. The<br />
set of $\kappa$-branches is denoted $[T]$. A $\kappa$-tree<br />
is an ''Aronszajn'' tree if it has no $\kappa$-branches.<br />
A cardinal $\kappa$ has the ''tree property'' if every<br />
$\kappa$-tree has a $\kappa$-branch.<br />
<br />
A transitive set $M$ is a $\kappa$-model of set theory if<br />
$|M|=\kappa$, $M^{\lt\kappa}\subset M$ and $M$ satisfies ZFC$^-$,<br />
the theory ZFC without the power set axiom (and using collection and separation rather than merely replacement). <br />
For any<br />
infinite cardinal $\kappa$ we have<br />
that $H_{\kappa^+}$ models ZFC$^-$, and further, if<br />
$M\prec H_{\kappa^+}$ and $\kappa\subset M$, then $M$ is<br />
transitive. Thus, any $A\in H_{\kappa^+}$ can be placed<br />
into such an $M$. If $\kappa^{\lt\kappa}=\kappa$, one can use<br />
the downward L&ouml;wenheim-Skolem theorem to find such $M$<br />
with $M^{\lt\kappa}\subset M$. So in this case there are abundant<br />
$\kappa$-models of set theory (and conversely, if there is<br />
a $\kappa$-model of set theory, then $2^{\lt\kappa}=\kappa$).<br />
<br />
The partition property $\kappa\to(\lambda)^n_\gamma$<br />
asserts that for every function $F:[\kappa]^n\to\gamma$<br />
there is $H\subset\kappa$ with $|H|=\lambda$ such that<br />
$F\upharpoonright[H]^n$ is constant. If one thinks of $F$ as<br />
coloring the $n$-tuples, the partition property asserts the<br />
existence of a ''monochromatic'' set $H$, since all<br />
tuples from $H$ get the same color. The partition property<br />
$\kappa\to(\kappa)^2_2$ asserts that every partition of<br />
$[\kappa]^2$ into two sets admits a set $H\subset\kappa$ of<br />
size $\kappa$ such that $[H]^2$ lies on one side of the<br />
partition. When defining $F:[\kappa]^n\to\gamma$, we define<br />
$F(\alpha_1,\ldots,\alpha_n)$ only when<br />
$\alpha_1<\cdots<\alpha_n$.<br />
<br />
== Weakly compact cardinals and the constructible universe ==<br />
<br />
Every weakly compact cardinal is weakly compact in [[Constructible universe|$L$]]. <cite>Jech2003:SetTheory</cite><br />
<br />
Nevertheless, the weak compactness property is not generally downward absolute between transitive models of set theory. <br />
<br />
<br />
== Weakly compact cardinals and forcing ==<br />
<br />
* Weakly compact cardinals are invariant under small forcing. [http://www.math.csi.cuny.edu/~fuchs/IndestructibleWeakCompactness.pdf]<br />
* Weakly compact cardinals are preserved by the canonical forcing of the GCH, by fast function forcing and many other forcing notions {{Citation needed}}.<br />
* If $\kappa$ is weakly compact, there is a forcing extension in which $\kappa$ remains weakly compact and $2^\kappa\gt\kappa$ {{Citation needed}}. <br />
* If the existence of weakly compact cardinals is consistent with ZFC, then there is a model of ZFC in which $\kappa$ is not weakly compact, but becomes weakly compact in a forcing extension <CITE>Kunen1978:SaturatedIdeals</CITE>.<br />
<br />
== Indestructibility of a weakly compact cardinal ==<br />
''To expand using [https://arxiv.org/abs/math/9907046]''<br />
<br />
== Relations with other large cardinals == <br />
<br />
* Every weakly compact cardinal is [[inaccessible]], [[Mahlo]], hyper-Mahlo, hyper-hyper-Mahlo and more. <br />
* [[Measurable]] cardinals, [[Ramsey]] cardinals, and [[indescribable|totally indescribable]] cardinals are all weakly compact and a stationary limit of weakly compact cardinals.<br />
* Assuming the consistency of a [[strongly unfoldable]] cardinal with ZFC, it is also consistent for the least weakly compact cardinal to be the least [[unfoldable]] cardinal. <cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite><br />
*If GCH holds, then the least weakly compact cardinal is not [[weakly measurable]]. However, if there is a [[measurable]] cardinal, then it is consistent for the least weakly compact cardinal to be weakly measurable. <cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite><br />
*If there is a $\kappa$ which is [[nearly supercompact|nearly $\theta$-supercompact]] where $\theta^{<\kappa}=\theta$, then it is consistent for the least weakly compact cardinal to be nearly $\theta$-supercompact. <cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite><br />
<br />
{{References}}</div>Wabb2thttp://cantorsattic.info/index.php?title=Measurable&diff=2048Measurable2017-11-11T19:04:25Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE: Measurable cardinal}}<br />
A '''measurable cardinal''' $\kappa$ is an [[uncountable]] [[cardinal]] such that it is possible to "measure" the subsets of $\kappa$ using a 2-valued [[measure]] on the powerset of $\kappa$, $\mathcal{P}(\kappa)$. There exists several other equivalent definitions: For example, $\kappa$ can also be the critical point of a nontrivial [[elementary embedding]] $j:V\to M$.<br />
<br />
Every measurable is a large cardinal, i.e. $V_\kappa$ satisfies ZFC, therefore ZFC cannot prove the existence of a measurable cardinal. In fact $\kappa$ is [[inaccessible]], the $\kappa$th inacessible, the $\kappa$th [[weakly compact]] cardinal, the $\kappa$th [[Ramsey]], and similarly bears most of the large cardinal properties under Ramsey-ness. It is notable that every measurable has the mentioned properties in ZFC, but in ZF they may not (but their existence remains consistency-wise ''much'' stronger than existence of cardinals with those properties), in fact under the [[axiom of determinacy]], the first two uncountable cardinals, $\aleph_1$ and $\aleph_2$, are both measurable.<br />
<br />
Measurable cardinals were introduced by Stanislaw Ulam in 1930.<br />
<br />
== Definitions ==<br />
<br />
The following definitions are equivalent for every uncountable cardinal $\kappa$:<br />
# There exists a 2-valued measure on $\kappa$.<br />
# There exists a $\kappa$-complete (or even just $\sigma$-complete) nonprincipal ultrafilter on $\kappa$.<br />
# There exists a nontrivial elementary embedding $j:V\to M$ with $M$ a transitive class and such that $\kappa$ is the least ordinal moved (the ''critical point'').<br />
# There exists an ultrafilter $U$ on $\kappa$ such that the [[ultrapower]] $(Ult_U(V),\in_U)$ of the universe is well-founded and isn't isomorphic to $V$.<br />
<br />
The equivalence between the first two definition is due to the fact that if $\mu$ is a 2-valued measure on $\kappa$, then $U=\{X\subset\kappa|\mu(X)=1\}$ is a nonprincipal ultrafilter (since $\mu$ is 2-valued) and is also $\sigma$-complete because of $\mu$'s $\sigma$-additivity. Similarly, if $U$ is a $\sigma$-complete nonprincipal ultrafilter on $\kappa$, then $\mu:\mathcal{P}(\kappa)\to[0,1]$ defined by $\mu(X)=1$ whenever $X\in U$, $\mu(X)=0$ otherwise is a 2-valued measure on $\kappa$.<br />
<br />
To see that the third definition implies the first two, one can show that if $j:V\to M$ is a nontrivial elementary embedding, then the set $\mathcal{U}=\{x\subset\kappa|\kappa\in j(x)\})$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, and in fact a normal fine measure. To show the converse, one needs to use [[ultrapower|ultrapower embeddings]]: if $U$ is a nonprincipal $\kappa$-complete ultrafilter on $\kappa$, then the canonical ultrapower embedding $j:V\to Ult_U(V)$ is a nontrivial elementary embedding of the universe.<br />
<br />
The equivalence of the last definition with the other ones is simply due to the fact that the ultrapower $(Ult_U(V),\in_U)$ of the universe is well-founded if and only if $U$ is $\sigma$-complete, and is isomorphic to $V$ if and only if $U$ is principal.<br />
<br />
== Properties ==<br />
<br />
''See also: [[Ultrapower]]''<br />
<br />
If $\kappa$ is measurable, then it has a measure that take every value in [0,1]. Also there must be a normal fine measure on $\mathcal{P}_\kappa(\kappa)$.<br />
<br />
Every measurable cardinal is [[regular]], and (under AC) bears most large cardinal properties weaker than it. It is in particular $\Pi^2_1$-[[indescribable]]. However the least measurable cardinal is not $\Sigma^2_1$-indescribable. Independently of the truth of AC, the existence of a measurable cardinal implies the consistency of the existence of large cardinals with the said properties, even if that measurable is merely $\omega_1$.<br />
<br />
If $\kappa$ is measurable and $\lambda<\kappa$ then it cannot be true that $\kappa<2^\lambda$. Under AC this means that $\kappa$ is a strong limit (and since it is regular, it must be strongly inaccessible, hence it cannot be $\omega_1$).<br />
<br />
If there exists a measurable cardinal then [[zero sharp|$0^\#$]] exists, and therefore $V\neq L$. In fact, the [[zero sharp|sharp]] of every real number exists, and therefore $\mathbf{\Pi}^1_1$-[[axiom of determinacy|determinacy]] holds. Furthermore, assuming the axiom of determinacy, the cardinals $\omega_1$, $\omega_2$, $\omega_{\omega+1}$ and $\omega_{\omega+2}$ are measurable, also in $L(\mathbb{R})$ every regular cardinal smaller than [[theta|$\Theta$]] is measurable.<br />
<br />
Every measurable has the following reflection property: let $j:V\to M$ be a nontrivial elementary embedding with critical point $\kappa$. If $x\in V_\kappa$ and $M\models\varphi(\kappa,x)$ for some first-order formula $\varphi$, then the set of all ordinals $\alpha<\kappa$ such that $V\models\varphi(\alpha,x)$ is [[stationary]] in $\kappa$ and has the same measure as $\kappa$ itself by any 2-valued measure on $\kappa$.<br />
<br />
Measurability of $\kappa$ is equivalent with $\kappa$-strong compactness of $\kappa$, and also with $\kappa$-supercompactness of $\kappa$ (fragments of [[strongly compact | strong compactness]] and [[supercompact | supercompactness]] respectively.) It is also consistent with ZFC that the first measurable cardinal and the first [[strongly compact]] cardinal are equal.<br />
<br />
If a measurable $\kappa$ is such that there is $\kappa$ [[strongly compact]] cardinals below it, then it is strongly compact. If it is a limit of strongly compact cardinals, then it is strongly compact yet not [[supercompact]]. If a measurable $\kappa$ has infinitely many [[Woodin]] cardinals below it, then the axiom of determinacy holds in $L(\mathbb{R})$, also the [[axiom of projective determinacy]] holds.<br />
<br />
If $\kappa$ is measurable in a ground model, then it is measurable in any forcing extension of that ground model whose notion of forcing has cardinality strictly smaller than $\kappa$. Prikry showed however that every measurable can be collapsed to a cardinal of cofinality $\omega$ and no other cardinal is collapsed.<br />
<br />
=== Failure of GCH at a measurable ===<br />
<br />
Gitik proved that the following statements are equiconsistent:<br />
* The generalized continuum hypothesis fails at a measurable cardinal $\kappa$, i.e. $2^\kappa > \kappa^+$<br />
* The singular cardinal hypothesis fails, i.e. there is a strong limit singular $\kappa$ such that $2^\kappa > \kappa^+$<br />
* There is a measurable cardinal of [[Mitchell rank | Mitchell order]] $\kappa^{++}$, i.e. $o(\kappa)=\kappa^{++}$<br />
<br />
Thus violating GCH at a measurable (or violating the SCH at any strong limit cardinal) is strictly stronger consistency-wise than the existence of a measurable cardinal.<br />
<br />
However, if the generalized continuum hypothesis fails at a measurable, then it fails at $\kappa$ many cardinals below it.<br />
<br />
== Real-valued measurable cardinal ==<br />
<br />
A cardinal $\kappa$ is '''real-valued''' measurable if there exists a $\kappa$-additive measure on $\kappa$. The smallest cardinal $\kappa$ carrying a $\sigma$-additive 2-valued measure must also carry a $\kappa$-additive measure, and is therefore real-valued measurable, also it is strongly inaccessible under AC.<br />
<br />
If a real-valued measurable cardinal is not measurable, then it must be smaller than (or equal to) $2^{\aleph_0}$. [[Martin's axiom]] implies that the continuum is not real-valued measurable.<br />
<br />
Solovay showed that the existence of a measurable cardinal is equiconsistent with the existence of a real-valued measurable cardinal. More precisely, he showed that if there is a measurable then there is generic extension in which $\kappa=2^{\aleph_0}$ and $\kappa$ is real-valued measurable, and conversely if there exists a real-valued measurable then it is measurable in some model of ZFC.<br />
<br />
== See also ==<br />
* [[Ultrapower]]<br />
* [[Mitchell order]]<br />
* [[Axiom of determinacy]]<br />
* [[Strongly compact]] cardinal<br />
<br />
== Read more ==<br />
* Jech, Thomas - ''Set theory''<br />
<br />
* Bering A., Edgar - ''A brief introduction to measurable cardinals''</div>Wabb2thttp://cantorsattic.info/index.php?title=Woodin&diff=2047Woodin2017-11-11T19:01:17Z<p>Wabb2t: </p>
<hr />
<div>{{DISPLAYTITLE:Woodin cardinal}}<br />
'''Woodin cardinals''' (named after W. Hugh Woodin) are a generalization of the notion of strong cardinals and have been used to calibrate the exact proof-theoretic strength of the [[axiom of determinacy]]. They can also be seen as weakenings of ''Shelah cardinals'', defined below. Their exact definition has several equivalent but different characterizations, each of which is somewhat technical in nature. Nevertheless, an inner model theory encapsulating infinitely many Woodin cardinals and slightly beyond has been developed.<br />
<br />
== Definition and some properties ==<br />
<br />
We first introduce the concept of ''$\gamma$-strongness for $A$'': an ordinal $\kappa$ is ''$\gamma$-strong for $A$'' (or $\gamma$-$A$-strong) if there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\kappa$ such that $V_{\kappa+\gamma}\subseteq M$ and $A\cap V_{\kappa+\gamma} = j(A)\cap V_{\kappa+\gamma}$. Intuitively, $j$ preserves $A$.<br />
<br />
We also introduce ''Woodin-ness in $\delta$'': for an infinite ordinal $\delta$, a set $X\subseteq\delta$ is ''Woodin in $\delta$'' if for every function $f:\delta\to\delta$, there is an ordinal $\kappa\in X$ with $\{f(\beta)|\beta<\kappa\}\subseteq\kappa$, there exists a nontrivial [[elementary embedding]] $j:V\to M$ with critical point $\kappa$ such that $V_{j(f)(\kappa)}\subseteq M$.<br />
<br />
An [[inaccessible]] cardinal $\delta$ is '''Woodin''' if any of the following (equivalent) characterizations holds <cite>Kanamori2009:HigherInfinite</cite>:<br />
* For any set $A\subseteq V_\delta$, there exists a $\kappa<\delta$ that is $\gamma$-strong for $A$ for every $\gamma<\kappa$.<br />
* For any set $A\subseteq V_\delta$, the set $S=\{\kappa<\delta|\kappa$ is $\gamma$-strong for $A$ for every $\gamma<\kappa\}$ is [[stationary]] in $\delta$.<br />
* The set $F=\{X\subseteq \delta|\delta\setminus X$ is not ''Woodin in $\delta$''$\}$ is a proper [[filter]], the ''Woodin filter'' over $\delta$.<br />
* For every function $f:\delta\to\delta$ there exists $\kappa<\delta$ such that $\{f(\beta)|\beta\in\kappa\}\subseteq\kappa$ and there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\kappa$ such that $V_{j(f)(\delta)}\subseteq M$.<br />
<br />
Let $\delta$ be Woodin, $F$ be the Woodin filter over $\delta$, and $S=\{\kappa<\delta|\kappa$ is $\gamma$-strong for $A$ for every $\gamma<\kappa\}$. Then $F$ is normal and $S\in F$. <cite>Kanamori2009:HigherInfinite</cite> This implies every Woodin cardinal is [[Mahlo]] and preceeded by a stationary set of [[measurable]] cardinals. However, Woodin cardinals are not [[weakly compact]] as they are ''not'' $\Pi^1_1$-[[indescribable]].<br />
<br />
Woodin cardinals are weaker consistency-wise then [[superstrong]] cardinals. In fact, every superstrong is preceeded by a stationary set of Woodin cardinals.<br />
<br />
The existence of a Woodin cardinal implies the consistency of ZFC + "the [[filter|nonstationary ideal]] on $\omega_1$ is $\aleph_2$-saturared". [[Huge]] cardinals were first invented to prove the consistency of the existence of a $\aleph_2$-saturated ideal on $\omega_1$, but turned out to be stronger than required, as a Woodin is enough.<br />
<br />
== Shelah cardinals ==<br />
<br />
Shelah cardinals were introduced by Shelah and Woodin as a weakening of the necessary hypothesis required to show several regularity properties of sets of reals hold in the model $L(\mathbb{R})$ (e.g., every set of reals is Lebesgue measurable and has the property of Baire, etc...). In slightly more detail, Woodin had established that the [[axiom of determinacy]] (a hypothesis known to imply regularity properties for sets of reals) holds in $L(\mathbb{R})$ <!--(see [[constructible universe]])-->assuming the existence of a nontrivial elementary embedding $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$ with critical point $<\lambda$. This axiom, a [[rank-into-rank]] axiom, is known to be very strong and its use was first weakened to that of the existence of a [[supercompact]] cardinal. Following the work of Foreman, Magidor and Shelah on saturated ideals on $\omega_1$, Woodin and Shelah subsequently isolated the two large cardinal hypotheses which bear their name and turn out to be sufficient to establish the [[projective#Regularity properties|regularity properties]] of sets of reals mentioned above.<br />
<br />
Shelah cardinals were the first cardinals to be devised by Woodin and Shelah. A cardinal $\delta$ is ''Shelah'' if for every function $f:\delta\to\delta$ there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\delta$ such that $V_{j(f)(\delta)}\subseteq M$. Every Shelah is Woodin, but not every Woodin is Shelah: indeed, Shelah cardinals are always measurable and in fact [[strong]], while Woodins are usually not. However, just like Woodins, Shelah cardinals are weaker consistency-wise than superstrong cardinals.<br />
<br />
A related notion is ''Shelah-for-supercompactness'', where the closure condition $V_{j(f)(\delta)}\subseteq M$ is replaced by $M^{j(f)(\delta)}\subseteq M$, a much stronger condition. The difference between Shelah and Shelah-for-supercompactness cardinals is essentially the same as the difference between strong and [[supercompact]] cardinals, or between [[superstrong]] and [[huge]] cardinals. Also, just like every Shelah is preceeded by a stationary set of strong cardinals, every Shelah-for-supercompactness cardinal is preceeded by a stationary set of supercompact cardinals. ''Woodin-for-supercompactness'' cardinals were also considered, but they turned out to be equivalent to [[Vopenka|Vopěnka]] cardinals.<br />
<br />
== Woodin cardinals and determinacy ==<br />
<br />
''See also: [[axiom of determinacy]], [[projective#Projective determinacy|axiom of projective determinacy]]''<br />
<br />
Woodin cardinals are linked to different forms of the [[axiom of determinacy]] <cite>Kanamori2009:HigherInfinite</cite><cite>Larson2010:HistoryDeterminacy</cite><cite>KoellnerWoodin2010:LCFD</cite>:<br />
* ZF+AD, ZFC+AD$^{L(\mathbb{R})}$, ZFC+"the non-stationary ideal over $\omega_1$ is $\omega_1$-dense" and ZFC+"there exists infinitely many Woodin cardinals" are equiconsistent.<br />
* Under ZF+AD, the model HOD$^{L(\mathbb{R})}$ satisfies ZFC+"$\Theta^{L(\mathbb{R})}$ is a Woodin cardinal". <cite>KoellnerWoodin2010:LCFD</cite> gives many generalization of this result.<br />
* If there exists infinitely many Woodin cardinals with a measurable above them all, then $L(\mathbb{R})$-determinacy. If there assumtion that there is a measurable above those Woodins is removed, one still has projective determinacy.<br />
* In fact projective determinacy is equivalent to "for every $n<\omega$, there is a fine-structural, countably iterable inner model $M$ such that $M$ satisfies ZFC+"there exists $n$ Woodin cardinals".<br />
* For every $n$, if there exists $n$ Woodin cardinals with a measurable above them all, then all $\mathbf{\Sigma}^1_{n+1}$ sets are determined.<br />
* $\mathbf{\Pi}^1_2$-determinacy is equivalent to "for every $x\in\mathbb{R}$, there is a countable ordinal $\delta$ such that $\delta$ is a Woodin cardinal in some inner model of ZFC containing $x$.<br />
* $\mathbf{\Delta}^1_2$-determinacy is equivalent to "for every $x\in\mathbb{R}$, there is an inner model M such that $x\in M$ and $M$ satisfies ZFC+"there is a Woodin cardinal". <br />
* ZFC + ''lightface'' $\Delta^1_2$-determinacy implies that there many $x$ such that HOD$^{L[x]}$ satisfies ZFC+"$\omega_2^{L[x]}$ is a Woodin cardinal".<br />
* Z$_2$+ $\Delta^1_2$-determinacy is conjectured to be equiconsistent with ZFC+"Ord is Woodin", where "Ord is Woodin" is expressed as an axiom scheme and Z$_2$ is [[:wikipedia:second-order arithmetic|second-order arithmetic]].<br />
* Z$_3$+ $\Delta^1_2$-determinacy is provably equiconsistent with NBG+"Ord is Woodin" where NBG is [[:wikipedia:Von Neumann–Bernays–Gödel set theory|Von Neumann–Bernays–Gödel set theory]] and Z$_3$ is third-order arithmetic.<br />
<br />
== Role in $\Omega$-logic ==<br />
== Stationary tower forcing ==<br />
<br />
{{References}}</div>Wabb2t