Constructible Hierarchy

Let be Constructible Sets from .
Define now

Note that is a Hierarchy.
Constructible Rank

Lemma

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

Proof

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

Lemma

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

Corollary

If is any Transitive set with and
where is Sufficiently Strong for Absoluteness of
then so .
In particular:

If contains all Ordinals, then .

Theorem