A class of formulas in a Language is
closed under bounded quantification
if for any :
Lemma
Let be a Transitive Model of .
Suppose is Absolute for .
Then and are Absolute for .
Proof
Let have Free Variables, one of which is
(otherwise we can’t quantify over )
By definition, we have for any
where is a Model of
Let .
Suppose
Then for some we have
and by assumption
is a Substructure of so (in ) and thus
Similar to .
Suppose
Then pick an such that and .
As by assumption, and is Transitive,
it has to be that and thus
(because is Absolute for )
Hence we are done as
Similar to .