http://cantorsattic.info/api.php?action=feedcontributions&user=Lok739&feedformat=atomCantor's Attic - User contributions [en]2022-08-18T23:26:09ZUser contributionsMediaWiki 1.24.4http://cantorsattic.info/index.php?title=Small_countable_ordinals&diff=2709Small countable ordinals2019-03-26T12:12:36Z<p>Lok739: </p>
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<div>{{DISPLAYTITLE:The small countable ordinals}}<br />
[[Category:Lower attic]]<br />
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The ordinals begin with the following transfinite progression<br />
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$0,1,2,3,\ldots,\omega,\omega+1,\omega+2,\omega+3,\ldots,\omega\cdot 2,\omega\cdot 2+1,\ldots,\omega\cdot 3,\ldots,\omega^2,\omega^2+1,\ldots,\omega^2+\omega,\ldots,\omega^2+\omega+1,\ldots,\omega^2+\omega\cdot 2,\ldots,\omega^3,\ldots,$<br />
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$\omega^\omega,\omega^\omega+1,\ldots,\omega^\omega+\omega,\ldots,\omega^\omega+\omega\cdot 2,\ldots,\omega^\omega\cdot 2,\ldots,\omega^{\omega^\omega},\ldots,\omega^{\omega^{\omega^\omega}},\ldots,\epsilon_{0}$<br />
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== Counting to $\omega^2$ == <br />
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We explain here in detail how to count to $\omega^2$. This is something that anyone can learn to do, even young children.<br />
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== The ordinals below [[epsilon naught | $\epsilon_0$]] == <br />
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We shall give here an account of the attractive finitary represenation of the ordinals below [[epsilon naught | $\epsilon_0$]].</div>Lok739http://cantorsattic.info/index.php?title=Elementary_embedding&diff=2708Elementary embedding2019-03-24T14:49:00Z<p>Lok739: </p>
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<div>Given two transitive structures $\mathcal{M}$ and $\mathcal{N}$, an '''elementary embedding''' from $\mathcal{M}$ to $\mathcal{N}$ is a function $j:\mathcal{M}\to\mathcal{N}$ such that $j(\mathcal{M})$ is an ''elementary substructure'' of $\mathcal{N}$, i.e. satisfies the same first-order sentences as $\mathcal{N}$ does. Obviously, if $\mathcal{M}=\mathcal{N}$, then $j(x)=x$ is an elementary embedding from $\mathcal{M}$ to itself, but is then called a '''trivial''' embedding. An embedding is '''nontrivial''' if there exists $x\in\mathcal{M}$ such that $j(x)\neq x$.<br />
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The critical point is the smallest ordinal moved by $j$. By $j$'s elementarity, $j(\kappa)$ must also be an ordinal, and therefore it is comparable with $\kappa$. It is easy to see why $j(\kappa)\leq\kappa$ is impossible, thus $j(\kappa)>\kappa$.<br />
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== Definition ==<br />
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Given two transitive classes $\mathcal{M}$ and $\mathcal{N}$, and a function $j:\mathcal{M}\rightarrow\mathcal{N}$, $j$ is an elementary embedding if and only if for every first-order formula $\varphi$ with parameters $x_1,...,x_n\in\mathcal{N}$, one has: $$\mathcal{M}\models\varphi(x_1,...,x_2)\iff\mathcal{N}\models\varphi(j(x_1),...,j(x_2))$$<br />
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$j$ is nontrivial if and only if it has a critical point, i.e. $\exists\kappa(j(\kappa)\neq\kappa)$. Indeed, one can show by transfinite induction that if $j$ does not move any ordinal then $j$ does not move any set at all, thus a critical point must exists whenever $j$ is nontrivial.<br />
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== Tarski-Vaught Test ==<br />
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If $\mathcal{M}$ and $\mathcal{N}$ are both $\tau$-structures for some language $\tau$, and $j:\mathcal{M}\rightarrow\mathcal{N}$, then $j$ is an elementary embedding iff:<br />
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#$j$ is injective (for any $x$ in $N$, there is ''at most'' one $y$ in $M$ such that $j(y)=x$). <br />
#$j$ has the following properties:<br />
##For any constant symbol $c\in\tau$, $j(c^\mathcal{M})=c^\mathcal{N}$.<br />
##For any function symbol $f\in\tau$ and $a_0,a_1...\in M$, $j(f^\mathcal{M}(a_0,a_1...))=f^\mathcal{N}(j(a_0),j(a_1)...)$. For example, $j(a_0+^\mathcal{M}a_1)=j(a_0)+^\mathcal{N}j(a_1)$.<br />
##For any relation symbol $r\in\tau$ and $a_0,a_1...\in M$, $r^\mathcal{M}(a_0,a_1...)\Leftrightarrow r^\mathcal{N}(j(a_0),j(a_1)...)$<br />
#For any first-order formula $\psi$ and any $x_0,x_1...\in M$ such that there is $y\in N$ with $\psi^\mathcal{N}(y,j(x_0),j(x_1)...)$, there is $z\in M$ with $\psi^\mathcal{M}(z,x_0,x_1...)$.<br />
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The '''Tarski-Vaught test''' refers to the special case where $\mathcal{M}$ is a substructure of $\mathcal{N}$ and $j(x)=x$ for every $x$. <br />
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This test determines if $\mathcal{M}$ is an elementary substructure of $\mathcal{N}$. More specifically, $\mathcal{M}$ is an elementary substructure of $\mathcal{N}$ iff for any $\psi$ and $x_0,x_1...\in M$ such that there is $y\in N$ with $\psi^\mathcal{N}(y,x_0,x_1...)$, there is $z\in M$ with $\psi^\mathcal{M}(z,x_0,x_1...)$. <br />
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== Use in Large Cardinal Axioms ==<br />
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There are two ways of making the critical point as large as possible:<br />
# Making $\mathcal{M}$ as large as possible, much larger than $\mathcal{N}$ (meaning that a "large" class can be embedded into a smaller class)<br />
# Making $\mathcal{M}$ and $\mathcal{N}$ more similar (for example, $\mathcal{M} = \mathcal{N}$ yet $j$ is nontrivial)<br />
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Using the first method, one can simply take $\mathcal{M}=V$ (the universe of all sets), and the resulting critical point is always a measurable cardinal, a very strong type of large cardinal, e.g. the first measurable is larger than infinitely many weakly compact cardinals (and much more).<br />
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Using the second method, one can take, say, $\mathcal{M} = \mathcal{N} = L$, i.e. create an embedding $j:L\to L$, whose existence has very important consequences, such as the existence of [[zero sharp|$0^\#$]] (and thus $V\neq L$) and implies that every ordinal that is an uncountable cardinal in V is strongly inaccessible in L. By taking $\mathcal{M}=\mathcal{N}=V_\lambda$, i.e. a rank of the cumulative hiearchy, one obtains the very powerful [[rank-into-rank]] axioms, which sit near the very top of the large cardinal hiearchy. However, this second method has its limits, as shown by Kunen, as he showed that $\mathcal{M}=\mathcal{N}=V$ leads to an inconsistency with the [[axiom of choice]], a theorem now known as the '''[[Kunen inconsistency]]'''. He also showed that a natural strengthening of the rank-into-rank axioms, $\mathcal{M}=\mathcal{N}=V_{\lambda+2}$ for some $\lambda\in Ord$, was inconsistent with the $AC$.<br />
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Most large cardinal axioms inbetween measurables and rank-into-rank axioms are obtained by mixing those two methods: one usually sets $\mathcal{M}=V$ then requires $\mathcal{N}$ to satisfies strong closure properties to make it "larger", i.e. closer to $V$ (that is, to $\mathcal{M}$). For example, $j:V\to\mathcal{N}$ is nontrivial with critical point $\kappa$ and the cumulative hiearchy rank $V_{j(\kappa)}$ is a subset of $\mathcal{N}$ then $\kappa$ is [[superstrong]]; if $\mathcal{N}$ contains all sequences of elements of $\mathcal{N}$ of length $\lambda$ for some $\lambda>\kappa$ then $\kappa$ is $\lambda$-[[supercompact]], and so on.<br />
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The existence of a nontrivial elementary embedding $j:\mathcal{M}\to\mathcal{N}$ ''that is definable in $\mathcal{M}$'' implies that the critical point $\kappa$ of $j$ is [[measurable]] in $\mathcal{M}$ (not necessarily in $V$). Every measurable ordinal is [[weakly compact]] and (strongly) [[inaccessible]] therefore its existence in any model is beyond $ZFC$, meaning that $ZFC$ cannot prove that such an embedding exists. <br />
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Here are some types of cardinals whose definition uses elementary embeddings:<br />
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* [[Reinhardt]] cardinals, [[Berkeley]] cardinals<br />
*[[Rank into rank]] cardinals (axioms I3-I0)<br />
*Huge cardinals: [[huge|almost n-huge]], [[huge|almost super-n-huge]], [[huge|n-huge]], [[huge|super-n-huge]], [[huge|$\omega$-huge]]<br />
*High jump cardinals: [[high-jump|almost high-jump]], [[high-jump]], [[high-jump|super high-jump]], [[high-jump|high-jump with unbounded excess closure]]<br />
*[[Extendible]] cardinals, [[extendible | $\alpha$-extendible]]<br />
*Compact cardinals: [[supercompact]], [[supercompact | $\lambda$-supercompact]], [[strongly compact]], [[nearly supercompact]]<br />
*Strong cardinals: [[strong]], [[strong | $\theta$-strong]], [[superstrong]], [[superstrong|super-n-strong]]<br />
*[[Measurable]] cardinals, measurables of nontrivial [[Mitchell order]], [[Tall]] cardinals<br />
*[[Weakly compact]] cardinals<br />
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The [[wholeness axioms]] also asserts the existence of elementary embeddings, though the resulting larger cardinal has no particular name. [[Vopenka|VopÄ›nka's principle]] is about elementary embeddings of set models.</div>Lok739http://cantorsattic.info/index.php?title=Aleph&diff=2707Aleph2019-03-24T10:42:52Z<p>Lok739: /* The continuum hypothesis */</p>
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<div>{{DISPLAYTITLE: The aleph numbers, $\aleph_\alpha$}}<br />
[[Category:Middle attic]]<br />
The aleph function, denoted $\aleph$, provides a 1 to 1 correspondence between the [[ordinal]] and the [[cardinal]] numbers. In fact, it is the only order-isomorphism between the ordinals and cardinals, with respect to membership. It is a strictly [[monotone]] ordinal function which can be defined via transfinite recursion in the following manner:<br />
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:$\aleph_0 = \omega$<br />
:$\aleph_{n+1} = \bigcap \{ x \in \operatorname{On} : | \aleph_n | \lt |x| \}$<br />
:$\aleph_a = \bigcup_{x \in a} \aleph_x$ where $a$ is a limit [[ordinal]].<br />
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To translate the formalism, $\aleph_{n+1}$ is the smallest ordinal whose cardinality is greater than the previous aleph. $\aleph_a$ is the limit of the sequence $\{ \aleph_0 , \aleph_1 , \aleph_2 , \ldots \}$ until $\aleph_a$ is reached when $a$ is a limit ordinal.<br />
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$\aleph_0$ is the smallest [[infinite]] [[cardinal]]. <br />
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== Aleph one == <br />
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$\aleph_1$ is the first [[uncountable]] cardinal. <br />
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== The continuum hypothesis == <br />
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The ''continuum hypothesis'' is the assertion that the set of real numbers $\mathbb{R}$ have cardinality $\aleph_{1}$. G&ouml;del showed the consistency of this assertion with ZFC, while Cohen showed using [[forcing]] that if ZFC is consistent then ZFC+$\aleph_1<|\mathbb R|$ is consistent.<br />
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== Equivalent Forms ==<br />
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The cardinality of the power set of $\aleph_{0}$ is $\aleph_{1}$<br />
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The is no set with cardinality $\alpha$ such that $\aleph_{0} < \alpha < \aleph_{1}$<br />
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== Generalizations ==<br />
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The ''generalized continuum hypothesis'' (GCH) states that if an infinite set's cardinality lies between that of an infinite set ''S'' and that of the [[power set]] of ''S'', then it either has the same cardinality as the set ''S'' or the same cardinality as the power set of ''S''. That is, for any [[infinite set|infinite]] [[cardinal number|cardinal]] <math>\lambda</math> there is no cardinal <math>\kappa</math> such that <math>\lambda <\kappa <2^{\lambda}.</math> GCH is equivalent to:<br />
:<math>\aleph_{\alpha+1}=2^{\aleph_\alpha}</math> for every [[ordinal number|ordinal]] <math>\alpha.</math> (occasionally called '''Cantor's aleph hypothesis''')<br />
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For more,see https://en.wikipedia.org/wiki/Continuum_hypothesis<br />
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== Aleph two ==<br />
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$\aleph_2$ is the second [[uncountable]] [[cardinal]]. <br />
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== Aleph hierarchy ==<br />
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The $\aleph_\alpha$ hierarchy of cardinals is defined by transfinite recursion:<br />
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* $\aleph_0$ is the smallest infinite cardinal. <br />
* $\aleph_{\alpha+1}=\aleph_\alpha^+$, the [[successor]] cardinal to $\aleph_\alpha$. <br />
* $\aleph_\lambda=\sup_{\alpha\lt\lambda}\aleph_\alpha$ for [[limit ordinal | limit ordinals]] $\lambda$. <br />
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Thus, $\aleph_\alpha$ is the $\alpha^{\rm th}$ infinite cardinal. In ZFC the sequence <br />
$$\aleph_0, \aleph_1,\aleph_2,\ldots,\aleph_\omega,\aleph_{\omega+1},\ldots,\aleph_\alpha,\ldots$$<br />
is an exhaustive list of all infinite cardinalities. Every infinite set is bijective with some $\aleph_\alpha$. <br />
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== Aleph omega ==<br />
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The cardinal $\aleph_\omega$ is the smallest instance of an [[uncountable]] [[singular]] [[cardinal]] number, since it is larger than every $\aleph_n$, but is the supremum of the [[countable]] set $\{\aleph_0,\aleph_1,\ldots,\aleph_n,\ldots\mid n\lt\omega\}$. <br />
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== Aleph fixed point == <br />
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A cardinal $\kappa$ is an ''$\aleph$-fixed point when $\kappa=\aleph_\kappa$. In this case, $\kappa$ is the $\kappa^{\rm th}$ infinite cardinal. Every [[inaccessible]] cardinal is an $\aleph$-fixed point, and a limit of such fixed points and so on. Indeed, every [[worldly]] cardinal is an $\aleph$-fixed point and a limit of such.<br />
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One may easily construct an $\aleph$-fixed point above any ordinal $\beta$: simply let $\beta_0=\beta$ and $\beta_{n+1}=\aleph_{\beta_n}$; it follows that $\kappa=\sup_n\beta_n=\aleph_{\aleph_{\aleph_{\aleph_{\ddots}}}}$ is an $\aleph$-fixed point, since $\aleph_\kappa=\sup_{\alpha\lt\kappa}\aleph_\alpha=\sup_n\aleph_{\beta_n}=\sup_n\beta_{n+1}=\kappa$. By continuing the recursion to any ordinal, one may construct $\aleph$-fixed points of any desired cofinality. Indeed, the class of $\aleph$-fixed points forms a closed unbounded class of cardinals.</div>Lok739