# 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. ## 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\}) (the transitive closure of \{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 \text{AC} (the axiom of choice). 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}. ## 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^{\#} exists (see below), then every uncountable cardinal \kappa has L\models"\kappais 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^{\#} exists. ## Statements True in L Here is a list of statements true in L: • \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 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.:[1] • 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. ## Sharps 0^{\#} (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^\# as a (real number encoding a) set of formulas, 0^\# provably exists in \text{ZFC}, but lacks all its important properties. Thus the expression "0^\# exists" is to be understood as "0^\# exists and there are uncountably many Silver indiscernibles". ### Definition of 0^{\#} Assume there is an uncountable set of Silver indiscernibles. Then 0^{\#} 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^{\#} 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^{\#} 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^{\#} does not exist" if there are no Silver indiscernibles. ### Implications, equivalences, and consequences of 0^\#'s existence If 0^\# exists then: • L_{\aleph_\omega}\prec L and so 0^\# 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 and totally ineffable and has most other virtual large cardinal properties and other large cardinal properties consistent with V=L.[1] • There are only countably many reals in L, i.e. |\R\cap L|=\aleph_0 in V. The following statements are equivalent: • There is an uncountable set of Silver indiscernibles (i.e. "0^\# exists") • There is a proper class of Silver indiscernibles (unboundedly many of them). • 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 (hence weakly inaccessible) in L. • There is a nontrivial elementary embedding j:L\to L. • There is a proper class of nontrivial elementary embeddings j:L\to L. • There is a nontrivial elementary embedding j:L_\alpha\to L_\beta with \text{crit}(j)<|\alpha|. The existence of 0^\# 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. Note that if 0^{\#} 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. Thus if any such embedding exists, then a proper class of those embeddings exists. ### Nonexistence of 0^\#, Jensen's Covering Theorem ### EM blueprints and alternative characterizations of 0^\# 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: 1. 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)) 2. 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. 3. 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\}$$0^\# is then defined as the unique EM blueprint T such that: 1. \mathcal{M}(T,\alpha) is isomorphic to a transitive model M(T,\alpha) of ZFC for every \alpha 2. For any infinite \alpha, the set of indiscernibles X associated with M(T,\alpha) can be made cofinal in \text{Ord}^{M(T,\alpha)}. 3. The L_\delta-indiscernables \beta_0<\beta_1... can be made so that if < is an M(T,\alpha)-definable well-ordering of M(T,\alpha), then for any (m+n+2)-ary formula \varphi such that \min_<^{M(T,\alpha)}\{x:\varphi[x,\beta_0,\beta_1...\beta_{m+n}]\}<\beta_m, then:$$\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$0^\#$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$L$.$0^\#$exists interestingly iff some$L_\delta$has an uncountable set of indiscernables. If$0^\#$exists, then there is some uncountable$\delta$such that$M(0^\#,\omega_1)=L_\delta$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$0^\#$. ### Sharps of arbitrary sets ### Generalisations$0^\dagger$(zero dagger) is a set of integers analogous to$0^\sharp$and connected with inner models of measurability.[2]$0^{sword}$is connected with nontrivial Mitchell rank.$¬ 0 ^{sword}$(not zero sword) means that there is no mouse with a measure of Mitchell order$> 0$.[3]$0^\P$(zero pistol) is connected with strong cardinals.$¬ 0^\P$(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$\mathrm{Ord}$).[3]$0^¶$is “the sharp for a strong cardinal”, meaning the minimal sound active mouse$\mathcal{M}$with$M | \mathrm{crit}(\dot F^{\mathcal{M}}) \models \text{“There exists a strong cardinaly”}$, with$\dot F^{\mathcal{M}}$being the top extender of$\mathcal{M}$.[4] ## References • Jech, Thomas J. Set Theory (The 3rd Millennium Ed.). Springer, 2003. • user46667, Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem), URL (version: 2014-03-17): https://mathoverflow.net/q/156940 • 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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