Constructible Hierarchy

Let be Constructible Sets from a set . Define now

Note that is a Hierarchy. Constructible Rank

Lemma

is Absolute for Transitive Models of a Sufficiently Strong i.e. there is a formula such that

This is Absolute for Transitive Models of .

Proof

The definition of is recursive.

Corollary

Let be Sufficiently Strong for Absoluteness of (above). Let be a Transitive Model of . Then

Proof

Let . Then we can show that . As is Transitive, we know that .

Lemma

If then .

Proof

By induction on . Clearly is countable. Also for all . Assume that and write

Suppose now is a limit and for , . Then write

Lemma

For we have where is the Von Neumann Hierarchy. For then .

Proof

First bit by -induction. Then . However, is countable and is not.