Constructible universe

Description

For a set $X$, let $\operatorname{Def}(X)$ be the set of all subsets of $X$ that are definable in the structure $(X, \in)$. Here a definition is a first-order formula with parameters from $X$. Define sets $L_\alpha$ by recursion on the ordinals $\alpha$: $$L_0 = \emptyset, \qquad L_{\alpha + 1} = \operatorname{Def}(L_\alpha), \qquad L_\lambda = \bigcup_{\alpha < \lambda} L_\alpha \text{ for a limit ordinal } \lambda.$$ The constructible universe is the class $L = \bigcup_{\alpha} L_\alpha$, where $\alpha$ runs over all ordinals. A set is constructible when it lies in $L$.

The class $L$ is an inner model of ZF. Each $L_\alpha$ is transitive, so $L$ is transitive. The ordinal $\alpha$ lies in $L_{\alpha + 1}$, because it is the set of ordinals of $L_\alpha$, and this set is definable over $L_\alpha$. So $L$ contains every ordinal. The axioms of ZF hold in $L$; the hardest case is the separation schema, which uses a reflection argument.