Let be the Language of set theory. We define the following

The smallest class of formulae which

A formula is if it is of the form

for

A formula is if it is of the form

for

A formula is if it is both and .

Definition

If is any Theory, and a class of formulae, define to be the class of formulas that are equivalent in to a formula in .

Theorem

formulas are Absolute for Transitive Models of .

Proof

By induction on the formula complexity in we prove the statement for . Note that the Transitiveness comes into play when dealing with bounded quantifiers. It is then straightforward to extend the statement for .

Corollary

formulas are Upwards Absolute for Transitive Models of , while are Downwards Absolute.

Corollary

formulas are Absolute for Transitive Models of .