Absolute Operation

Let be the Language of set theory, and , be -Structures We say that a Definable Operation is absolute for if it’s definable by and is Absolute between and

Lemma

If and are absolute for then so is .

Lemma

If is an absolute operation and is Absolute for a Transitive Model of then so are

and

Proof

First formula is equivalent to

and is Absolute, and is bounded, so we are good, as we are Closed Under Bounded Quantification. The second formula is equivalent to

This is unbounded, but also equivalent to

and thus both of them are Absolute by the -trick.

Theorem

Any arithmetic function is absolute.

Proof

Using the previous lemmas and Absoluteness Of Recursive Operations.