## § 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.