Extension of a transitive set
In model theory and set theory, which are disciplines within mathematics, a model
of some axiom system of set theory
in the language of set theory is an end extension of
, in symbols
, if
is a substructure of
, (i.e.,
and
), and
whenever
and
hold, i.e., no new elements are added by
to the elements of
.[1]
The second condition can be equivalently written as
for all
.
For example,
is an end extension of
if
and
are transitive sets, and
.
A related concept is that of a top extension (also known as rank extension), where a model
is a top extension of a model
if
and for all
and
, we have
, where
denotes the rank of a set.