Constructible universe

In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by ${\displaystyle L,}$ is a particular class of sets that can be described entirely in terms of simpler sets. ${\displaystyle L}$ is the union of the constructible hierarchy ${\displaystyle L_{\alpha }}$. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this paper, he proved that the constructible universe is an inner model of ZF set theory (that is, of Zermelo–Fraenkel set theory with the axiom of choice excluded), and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.

${\displaystyle L}$ can be thought of as being built in "stages" resembling the construction of the von Neumann universe, ${\displaystyle V}$. The stages are indexed by ordinals. In von Neumann's universe, at a successor stage, one takes ${\displaystyle V_{\alpha +1}}$ to be the set of all subsets of the previous stage, ${\displaystyle V_{\alpha }}$. By contrast, in Gödel's constructible universe ${\displaystyle L}$, one uses only those subsets of the previous stage that are:

By limiting oneself to sets defined only in terms of what has already been constructed, one ensures that the resulting sets will be constructed in a way that is independent of the peculiarities of the surrounding model of set theory and contained in any such model.

Define the Def operator:

$${\displaystyle \operatorname {Def} (X):={\Bigl \{}\{y\mid y\in X{\text{ and }}(X,\in )\models \Phi (y,z_{1},\ldots ,z_{n})\}~{\Big |}~\Phi {\text{ is a first-order formula and }}z_{1},\ldots ,z_{n}\in X{\Bigr \}}.}$$

${\displaystyle L}$ is defined by transfinite recursion as follows:

If ${\displaystyle z}$ is an element of ${\displaystyle L_{\alpha }}$, then ${\displaystyle z=\{y\in L_{\alpha }\ {\text{and}}\ y\in z\}\in {\textrm {Def}}(L_{\alpha })=L_{\alpha +1}}$. So ${\displaystyle L_{\alpha }}$ is a subset of ${\displaystyle L_{\alpha +1}}$, which is a subset of the power set of ${\displaystyle L_{\alpha }}$. Consequently, this is a tower of nested transitive sets. But ${\displaystyle L}$ itself is a proper class.

The elements of ${\displaystyle L}$ are called "constructible" sets; and ${\displaystyle L}$ itself is the "constructible universe". The "axiom of constructibility", aka "${\displaystyle V=L}$", says that every set (of ${\displaystyle V}$) is constructible, i.e. in ${\displaystyle L}$.