Transitive Model

Suppose and are -Structures,
where is the Language of set theory i.e.
and
We say that is transitive in if
for any such that

• (in )
  implies that

Lemma

If is a set theoretic universe i.e.
then is transitive in
if and only if
is a Transitive set.

Lemma

Let be the set theoretic universe.
If is transitive in then Axiom of Extensionality + Axiom of Foundation

Proof

Extensionality

Let such that
By Axiom of Extensionality (in ) there is (WLOG) some
Now by Transitivity of we know and so
Thus

Foundation

Suppose .
Find using Axiom of Foundation an -minimal (in )
By Transitivity we get
We can check that is still -minimal in .