Constructible universe
The Constructible universe (denoted $L$) was invented by Kurt Gödel as a transitive inner model of $\text{ZFC+}$$\text{GCH}$ (assuming the consistency of $\text{ZFC}$) showing that $\text{ZFC}$ cannot disprove $\text{GCH}$. It was then shown to be an important model of $\text{ZFC}$ for its satisfying of other axioms, thus making them consistent with $\text{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 $\text{ZFC}$, and implies many axioms which are consistent with $\text{ZFC}$. A set $X$ is constructible iff $X\in L$. $V=L$ iff every set is constructible.
Contents
Definition
$\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:
- $L_0=\emptyset$
- $L_{\alpha+1}=\mathrm{def}(L_\alpha)$
- $L_\beta=\bigcup_{\alpha<\beta} L_\alpha$ if $\beta$ is a limit ordinal
- $L=\bigcup_{\alpha\in\mathrm{Ord}} L_\alpha$
The Relativized constructible universes $L_\alpha(W)$ and $L_\alpha[W]$
$L_\alpha(W)$ for a class $W$ is defined the same way except $L_0(W)=\text{TC}(\{W\})$, where TC denotes the transitive closure, here of the set containing only $W$.
$L_\alpha[W]$ for a class $W$ is defined in the same way as $L$ 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$.
$L[W]=\bigcup_{\alpha\in\mathrm{Ord}}L_\alpha[W]$ is always a model of $\text{ZFC}$, and always satisfies $\text{GCH}$ past a certain cardinality. $L(W)=\bigcup_{\alpha\in\mathrm{Ord}}L_\alpha(W) $is always a model of $\text{ZF}$ but need not satisfy the axiom of choice ($\text{AC}$). In particular, $L(\mathbb{R})$ is, under large cardinal assumptions, a model of the axiom of determinacy. However, Shelah proved that if $\lambda$ is a strong limit cardinal of uncountable cofinality then $L(\mathcal{P}(\lambda))$ is a model of $\text{AC}$.
For an ordinal $\alpha$ and a class $u$, the following two conditions are equivalent:
- There is a $\Sigma_1$-definable $a\subset u$ such that $a\notin L_\alpha[u]$.
- There is a $\Sigma_1$-definable map from a subset of $u$ onto $L_\alpha[u]$.
These two conditions are also equivalent:
- There is a $\Delta_1$-definable $a\subset u$ such that $a\notin L_\alpha[u]$.
- There is a $\Delta_1$-definable map from a subset of $u$ onto $L_\alpha[u]$.
Source: [1] (pp. 14-16)
The difference between $L_\alpha$ and $V_\alpha$
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$-fixed point, $|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$.
If $0^{\sharp}$ exists (see below), then every uncountable cardinal $\kappa$ has $L\models$"$\kappa$is totally ineffable (and therefore the smallest actually totally ineffable cardinal $\lambda$ has many more large cardinal properties in $L$).
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^{\sharp}$ exists.
Statements True in $L$
Here is a list of statements true in $L$ of any model of $\text{ZF}$:
- $\text{ZFC}$ (and therefore the Axiom of Choice)
- $\text{GCH}$
- $V=L$ (and therefore $V$ $=$ $\text{HOD}$)
- The diamond principle
- The clubsuit principle
- The falsity of Suslin's hypothesis
Determinacy of $L(\R)$
Main article: axiom of determinacy
Using other logic systems than first-order logic
When using second order logic in the definition of $\mathrm{def}$, the new hierarchy is called $L_\alpha^{II}$. Interestingly, $L^{II}=\text{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)$. Finally, when using $\mathcal{L}_{\infty,\infty}$, it turns out that the result is $V$.
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 $\text{ZFC}$ closed under sequences of length $<\kappa$.
Silver indiscernibles
$S$ is a class of Silver indiscernibles iff $S$ is a class of ordinals and for each uncountable cardinal $\kappa$ [1]^{(Definition 3.18)}
- $\kappa \in S$
- $S \cap \kappa$ has order type $\kappa$
- $S \cap \kappa$ is a club in $\kappa$ if $\kappa$ is regular
- $S \cap \kappa$ is a set of indiscernibles for $\langle L_\kappa, \in \rangle$
- $Hull^{L_\kappa}(S \cap \kappa)=L_\kappa$
To be expanded.
Silver cardinals
A cardinal $κ$ is Silver if in a set-forcing extension there is a club in $κ$ of generating indiscernibles for $V_κ$ of order-type $κ$. This is a very strong property downwards absolute to $L$, e.g.:[2]
- Every element of a club $C$ witnessing that $κ$ is a Silver cardinal is virtually rank-into-rank.
- If $C ∈ V[H]$, a forcing extension by $\mathrm{Coll}(ω, V_κ)$, is a club in $κ$ of generating indiscernibles for $V_κ$ of order-type $κ$, then each $ξ ∈ C$ is $< ω_1$-iterable.
Jensen's dichotomy
Ronald Jensen proved that one of these two conditions must be true, and they are mutually exclusive:[3]
- For singular cardinal $\gamma$, $\gamma$ is singular in $L$ and $L$ is correct about $\gamma^+$ (i.e. $(\gamma^+)^L=\gamma^+$)
- Every uncountable cardinal is inaccessible in $L$
Sharps
$0^{\sharp}$ (zero sharp) is a $\Sigma_3^1$ real number which, under the existence of many Silver indiscernibles (a statement independent of $\text{ZFC}$), has a certain number of properties that contredicts the axiom of constructibility and implies that, in short, $L$ and $V$ are "very different". Technically, under the standard definition of $0^\sharp$ as a (real number encoding a) set of formulas, $0^\sharp$ provably exists in $\text{ZFC}$, but lacks all its important properties. Thus the expression "$0^\sharp$ exists" is to be understood as "$0^\sharp$ exists and there are uncountably many Silver indiscernibles".
Definition of $0^{\sharp}$
Assume there is an uncountable set of Silver indiscernibles. Then $0^\sharp$ is defined as the set of all Gödel numberings of first-order formula $\varphi$ such that $L_{\aleph_{\omega}}\models\varphi(\aleph_0,\aleph_1...\aleph_n)$ for some $n$.
"$0^{\sharp}$ exists" is used as a shorthand for "there is an uncountable set of Silver indiscernibles"; since $L_{\aleph_\omega}$ is a set, $\text{ZFC}$ can define a truth predicate for it, and so the existence of $0^\sharp$ as a mere set of formulas would be trivial. It is interesting only when there are many (in fact proper class many) Silver indiscernibles. Similarly, we say that "$0^\sharp$ does not exist" if there are no Silver indiscernibles.
Implications, equivalences, and consequences of $0^\sharp$'s existence
If $0^\sharp$ exists then:
- $L_{\aleph_\omega}\prec L$ and so $0^\sharp$ also corresponds to the set of the Gödel numberings of first-order formulas $\varphi$ such that $L\models\varphi(\aleph_0,\aleph_1...\aleph_n)$
- In fact, $L_\kappa\prec L$ for every Silver indiscernible, and thus for every uncountable cardinal.
- 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).
- For every $\alpha\in\omega_1^L$, every Silver indiscernible (and in particular every uncountable cardinal) in $L$ is a Silver cardinal, $\alpha$-iterable, $\geq$ an $\alpha$-Erdős, totally ineffable and completely remarkable and has most other virtual large cardinal properties and other large cardinal properties consistent with $V=L$.[2][4]
- There are only countably many reals in $L$, i.e. $|\R\cap L|=\aleph_0$ in $V$.
- By elementary-embedding absoluteness results (The hypothesis can be weakened, because one can chop at off the universe at any Silver indiscernible and use reflection.):[5]
- $L$, equipped with only its definable classes, is a model of the generic Vopěnka principle.
- In $L$ there are numerous virtual rank-into-rank embeddings $j : V_\theta^L → V_\theta^L$, where $\theta$ is far above the supremum of the critical sequence.
- Therefore every Silver indiscernible
- is virtually $A$-extendible in $L$ for every definable class $A$
- and is the critical point of virtual rank-into-rank embeddings with targets as high as desired and fixed points as high above the critical sequence as desired.
- There is a class-forcing notion $\mathbb{P}$ definable in $L$, such that in any $L$-generic extension $L[C]$ by this forcing, $\text{GBC}$ and the generic Vopěnka principle hold, yet $\text{Ord}$ is not Mahlo.[5]
- Proof includes a lemma stating: For any ordinal $\delta$ and any natural number (of the meta-theory — this lemma is a scheme) $n$, if $D_{\delta,n} ⊂ \mathbb{P}$ is the collection of conditions $c$ for which there is an ordinal $\theta$ such that
- $L_\theta \prec_{\Sigma_n} L$,
- $c \cap \theta$ is $L_\theta$-generic for $\mathbb{P}^{L_\theta}$ and
- in some forcing extension of $L$, there is an elementary embedding
- $j : ⟨ L_\theta , \in, c \cap \theta ⟩ \to ⟨ L_\theta , \in, c \cap \theta⟩$
- with critical point above $\delta$,
- then $D_{\delta,n}$ is a definable dense subclass of $\mathbb{P}$ in $L$.
- Proof includes a lemma stating: For any ordinal $\delta$ and any natural number (of the meta-theory — this lemma is a scheme) $n$, if $D_{\delta,n} ⊂ \mathbb{P}$ is the collection of conditions $c$ for which there is an ordinal $\theta$ such that
- There is a definable class-forcing notion in $L$, such that in the corresponding $L$-generic extension, $\text{GBC}$ holds, the generic Vopěnka scheme holds, but $\text{Ord}$ is not definably Mahlo, because there is a $\Delta_2$-definable club class avoiding the regular cardinals.
- There is a class-forcing extension $L[G]$ of the constructible universe in which the generic Vopěnka principle holds (so $gVP(\kappa, \mathbf{\Sigma_{n+1}})$ and $gVP(\Pi_n)$ hold for any $\kappa$ and $n$), but there are no $\Sigma_2$-reflecting cardinals and hence no remarkable cardinals (or $n$-remarkable cardinals).[5]
The following statements are equivalent:
- There is an uncountable set of Silver indiscernibles (i.e. "$0^\sharp$ exists")
- There is a proper class of Silver indiscernibles (unboundedly many of them).
- There is some $L_\alpha$ containing uncountably many indiscernibles.[1] (see below)
- There is a unique well-founded remarkable E.M. set (see below).
- Jensen's Covering Theorem fails (see below).
- $L$ is thin, i.e. $|L\cap V_\alpha|=|\alpha|$ for all $\alpha\geq\omega$.
- $\Sigma^1_1$-determinacy (lightface form).
- $\aleph_\omega$ is regular (equivalently weakly inaccessible, equivalently strongly inaccessible) in $L$.
- Every uncountable cardinal is inaccessible in $L$ (part of Jensen's dichotomy).[3][6]
- There is a singular cardinal that is regular in $L$. (equivalence for this and the following is implied by Jensen's dichotomy )
- There is a singular cardinal $\gamma$ such that $L$ is not correct about $\gamma^+$ (i.e. $(\gamma^+)^L\neq\gamma^+$).
- There is an $L$-ultrafilter by which the ultrapower of $L$ is well-founded.[1]
- There is an iterable L-ultrafilter.[1]
- There is an active baby mouse.[1]
- There is a nontrivial elementary embedding $j:L\to L$.[7]^{(Theorem 3.2)}
- This statement cannot be stated in this form in ZFC and requires for $j$ to be sufficiently definable in $V$. Being a class (and actually much less) is enough, but in other sense it can be false.
- E.g., if universe $\langle V, \in, j \rangle$ is a model of ZFC + BTEE + "$0^\sharp$ does not exist" (such a model can be obtained assuming the existence of $\omega_1$-Erdős cardinal), then $\langle L, \in, j \upharpoonright L\rangle$, an inherited model obtained by restriction of $j$, is a counterexample (has $j : L \to L$, but no $0^\sharp$).^{(Example 9.2)}
- There is a proper class of nontrivial elementary embeddings $j:L\to L$.
- If $0^\sharp$ exists then for every Silver indiscernible (in particular for every uncountable cardinal) there is a nontrivial elementary embedding $j:L\rightarrow L$ with that indiscernible as its critical point.
- There is a nontrivial elementary embedding $j:L_\alpha\to L_\beta$ with $\mathrm{crit}\,j < |\alpha|$.[7]^{(Theorem 3.2)}
- For every uncountable cardinal $\kappa$, there is a nontrivial elementary embedding $j:L_\kappa\to L_\kappa$.[7]^{(Theorem 3.2)}
- There is a model $\langle M, \in, j \rangle$ that satisfies ZFC + BTEE, such that $M$ is a transitive class containing all the ordinals.[7]^{(Proposition 9.1)}
- There is a transitive set $M$ and an elementary embedding $j: M → M$ such that $\mathrm{Ord}^M$ is an uncountable cardinal.[7]^{(Proposition 9.1)}
The existence of $0^\sharp$ is implied by:
- Chang's conjecture
- Both $\omega_1$ and $\omega_2$ being singular (requires $\neg\text{AC}$).
- The negation of the singular cardinal hypothesis ($\text{SCH}$).
- The existence of an $\omega_1$-iterable cardinal or of a $\omega_1$-Erdős cardinal.
- The existence of a weakly compact cardinal $\kappa$ such that $|(\kappa^+)^L|=\kappa$.
- The existence of some uncountable regular cardinal $\kappa$ such that every constructible $X\subseteq\kappa$ either contains or is disjoint from a closed unbounded set.
Nonexistence of $0^\sharp$, Jensen's Covering Theorem
EM blueprints and alternative characterizations of $0^\sharp$
An EM blueprint (Ehrenfeucht-Mostowski blueprint) $T$ is any theory of the form $\{\varphi:(L_\delta;\in,\alpha_0,\alpha_1...)\models\varphi\}$ for some ordinal $\delta>\omega$ and $\alpha_0<\alpha_1<\alpha_2...$ are indiscernible in the structure $L_\delta$. Roughly speaking, it's the set of all true statements about $\alpha_0,\alpha_1,\alpha_2...$ in $L_\delta$.
For an EM blueprint $T=\{\varphi:(L_\delta;\in,\alpha_0,\alpha_1...)\models\varphi\}$, the theory $T^{-}$ is defined as $\{\varphi:L_\delta\models\varphi\}$ (the set of truths about any definable elements of $L_\delta$). Then, the structure $\mathcal{M}(T,\alpha)=(M(T,\alpha);E)\models T^{-}$ has a very technical definition, but it is indeed uniquely (up to isomorphism) the only structure which satisfies the existence of a set $X$ of $\mathcal{M}(T,\alpha)$-ordinals such that:
- $X$ is a set of indiscernibles for $\mathcal{M}(T,\alpha)$ and $(X;E)\cong\alpha$ ($X$ has order-type $\alpha$ with respect to $\mathcal{M}(T,\alpha)$)
- For any formula $\varphi$ and any $x<y<z...$ with $x,y,z...\in X$, $\mathcal{M}(T,\alpha)\models\varphi(x,y,z...)$ iff $\mathcal{M}(T,\alpha)\models\varphi(\alpha_0,\alpha_1,\alpha_2...)$ where $\alpha_0,\alpha_1...$ are the indiscernibles used in the EM blueprint.
- If $<$ is an $\mathcal{M}(T,\alpha)$-definable $\mathcal{M}(T,\alpha)$-well-ordering of $\mathcal{M}(T,\alpha)$, then: $$\mathcal{M}(T,\alpha)=\{\min{}_<^{\mathcal{M}(T,\alpha)}\{x:\mathcal{M}(T,\alpha)\models\varphi[x,a,b,c...]\}:\varphi\in\mathcal{L}_\in\text{ and } a,b,c...\in X\}$$
$0^\sharp$ is then defined as the unique EM blueprint $T$ such that:
- $\mathcal{M}(T,\alpha)$ is isomorphic to a transitive model $M(T,\alpha)$ of ZFC for every $\alpha$
- For any infinite $\alpha$, the set of indiscernibles $X$ associated with $M(T,\alpha)$ can be made cofinal in $\text{Ord}^{M(T,\alpha)}$.
- The $L_\delta$-indiscernables $\beta_0<\beta_1...$ can be made so that if $<$ is an $M(T,\alpha)$-definable well-ordering of $M(T,\alpha)$, then for any $(m+n+2)$-ary formula $\varphi$ such that $\min_<^{M(T,\alpha)}\{x:\varphi[x,\beta_0,\beta_1...\beta_{m+n}]\}<\beta_m$, then: $$\min{}_<^{M(T,\alpha)}\{x:\varphi[x,\beta_0,\beta_1...\beta_{m+n}]\}=\min{}_<^{M(T,\alpha)}\{x:\varphi[x,\beta_0,\beta_1...\beta_{m-1},\beta_{m+n+1}...\beta_{m+2n+1}]\}$$
If the EM blueprint meets 1. then it is called well-founded. If it meets 2. and 3. then it is called remarkable.
If $0^\sharp$ exists (i.e. there is a well-founded remarkable EM blueprint) then it happens to be equivalent to the set of all $\varphi$ such that $L\models\varphi[\kappa_0,\kappa_1...]$ for some uncountable cardinals $\kappa_0,\kappa_1...<\aleph_\omega$. This is because the associated $M(T,\alpha)$ will always have $M(T,\alpha)\prec L$ and furthermore $\kappa_0,\kappa_1...$ would be indiscernibles for $L$.
$0^\sharp$ exists interestingly iff some $L_\delta$ has an uncountable set of indiscernables. If $0^\sharp$ exists, then there is some uncountable $\delta$ such that $M(0^\sharp,\omega_1)=L_\delta$ and $L_\delta$ therefore has an uncountable set of indiscernables. On the other hand, if some $L_\delta$ has an uncountable set of indiscernables, then the EM blueprint of $L_\delta$ is $0^\sharp$.
Sharps of arbitrary sets
Definition: TODO
One can talk about $0^{\sharp\sharp}$[8] or $\mathbb{R}^\sharp$.
“$\forall_{a \in {}^\omega\omega} \text{$a^\sharp$ exists}$” is stronger than “$\text{$0^\sharp$ exists}$”, but weaker then an $\omega_1$-Erdős cardinal.[9]
The core model contains “all the sharps”.[8]
$V_\delta^{n\sharp}$ ($V_\delta^\sharp$, $V_\delta^{\sharp\sharp}$ etc.) are examples of possible Icarus sets strenghtening the $\mathrm{I0}$ axiom.[10, 11]
If $X^\sharp$ exists for every set $X$, then an axiom of generic absoluteness, $\mathcal{A}(H(\omega_1), \underset{\sim}{\Sigma_2})$, holds.[12]
Every set has a sharp if and only if every $\mathbf{\Sigma}^1_2$ set of reals is universally Baire.
Generalisations
$0^\dagger$ (zero dagger) is a set of integers analogous to $0^\sharp$ and connected with inner models of measurability.[13]
$BMM$ (bounded Martin’s maximum) implies that for every set $X$ there is an inner model with a strong cardinal containing $X$.[12]
- Thus, in particular, $BMM$ implies that for every set $X$, $X^\dagger$ exists.
$0^{sword}$ is connected with nontrivial Mitchell rank. $\lnot 0 ^{sword}$ (not zero sword) means that there is no mouse with a measure of Mitchell order $> 0$.[14]
$0^\P$ (zero pistol) is connected with strong cardinals. $\lnot 0^\P$ (not zero pistol) means that a core model may be built with a strong cardinal, but that there is no class of indiscernibles for it that is closed and unbounded in $\mathrm{Ord}$).[14] $0^¶$ is "the sharp for a strong cardinal", meaning the minimal sound active mouse $\mathcal{M}$ with $M | \mathrm{crit}(\dot F^{\mathcal{M}}) \models ``\textrm{There exists a strong cardinal}\!"$, with $\dot F^{\mathcal{M}}$ being the top extender of $\mathcal{M}$.[15]
References
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- 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
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- Nielsen, Dan Saattrup and Welch, Philip. Games and Ramsey-like cardinals. , 2018. arχiv bibtex