Suppose and are -Structures,
where is the Language of set theory i.e.
and (i.e. is a Substructure of )
We say that is a transitive substructure of if
for any :
We will often say is a transitive model of .
This means that is a transitive substructure of such that
where is the set theoretic universe.
Special case
is a transitive substructure of
if and only if
is Transitive.