§ The constructible universe L

• When building von neumann universe , we take all subsets from previous state; $V(0) = \emptyset$, $V(n + 1) = 2^{V(n)}$, $V(\lim \alpha) = \cup_{\beta < \alpha} V(\beta)$.
• To build $L$ (the definable universe), first we need the notion of definability.
• For a set $X$, the set $Def(X)$ is the set of all $Y \subseteq X$ such that $Y$ is logically definable in the structure $(X, \in)$ (That is, we are given access to FOL and $\in$) from parameters in $X$ (that is, we can have free variables of elements of $X$).
• We can now build the constructible universe by iteratively constructing definable sets of the previous level.
• Can talk about definability in terms of godel operations , which has ordered & unordered pairing, cartesian product, set difference, taking the domain of a binary relation, automorphisms of an ordered triple. These give us a "constructive" description of what we can do using definability. See also: constructible universe at nLab
• Computable universe

§ Godel Normal Form theorem

• Theorem which says that constructible sets are those that can be built from godel operations.