Constructible universe

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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 $L\models$"$\kappa$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 $0^{\#}$ 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.

Sharps

$0^{\#}$ 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) is $\alpha$-iterable, $\geq$ an $\alpha$-Erdős, and totally ineffable in $L$.
  • 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

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