§ The constructible universe L
- When building von neumann universe , we take all subsets from previous state; , , .
- To build (the definable universe), first we need the notion of definability.
- For a set , the set is the set of all such that is logically definable in the structure (That is, we are given access to FOL and ) from parameters in (that is, we can have free variables of elements of ).
- 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.