The Constructible Universe

Gödel's constructible universe $L$ is a canonical inner model of set theory in which every set is "constructible" from below using definable operations. It provides the foundation for consistency results such as the consistency of the Axiom of Choice and the Generalized Continuum Hypothesis with ZF.

Definition

Definition7.1Constructible Hierarchy

The constructible hierarchy is defined by transfinite recursion:

$L_0 = \emptyset$
$L_{\alpha+1} = \mathcal{Def}(L_\alpha)$ , the set of all subsets of $L_\alpha$ definable by first-order formulas with parameters from $L_\alpha$
$L_\lambda = \bigcup_{\alpha < \lambda} L_\alpha$ for limit ordinals $\lambda$
$L = \bigcup_{\alpha \in \mathrm{Ord}} L_\alpha$

A set $x$ is constructible if $x \in L$ .

Definition7.2Axiom of Constructibility

The axiom $V = L$ states that every set is constructible: $\forall x \, (x \in L)$ . This axiom is consistent with ZFC (if ZFC is consistent) and decides many set-theoretic questions left open by ZFC alone.

Properties

ExampleKey Properties of $L$

The constructible universe satisfies:

ZFC: $L \models \text{ZFC}$ (Gödel). In particular, $L \models \text{AC}$ .
GCH: $L \models 2^{\aleph_\alpha} = \aleph_{\alpha+1}$ for all ordinals $\alpha$ .
$\Diamond$ : $L$ satisfies Jensen's diamond principle, which implies the existence of Suslin trees and other combinatorial objects.
Condensation: The Condensation Lemma states that every elementary substructure of $L_\alpha$ is isomorphic to some $L_\beta$ .

RemarkMinimality of $L$

$L$ is the smallest inner model of ZF containing all ordinals. For any transitive class $M$ with $M \models \text{ZF}$ and $\mathrm{Ord} \subseteq M$ , we have $L \subseteq M$ . This makes $L$ a "canonical" model against which other set-theoretic universes can be compared.