Absolute

Let and be Structures for a Language such that
Let be a formula in Free Variables.
We say that is absolute between and if
it is Upwards Absolute and Downwards Absolute i.e.

where we use to mean is Satisfied in

Theorem

If is a Substructure of ,
then all quantifier free formulas are absolute between and

Theorem

If and are Absolute and

then both and are Absolute for Models of .
We call this the trick.

Proof

Let be a Model of and a Substructure of
such that
Firstly, is Upwards Absolute so is Upwards Absolute.
Suppose .
Then and thus
But then and thus is Upwards Absolute.
But is Downwards Absolute because is Downwards Absolute.
So is absolute and so is (by similar arguments).

(Non)example

Language of set theory has symbols
Take

meaning
(note in particular that cannot be quantifier free)
Also take

meaning

Take a model

Let be such that (exists by Empty-set axiom)
and such that (exists by Pair-set axiom …)
Then take
But then !!
and thus is not absolute in the Language of set theory.