In model theory and set theory, which are disciplines within mathematics, a model
B
=
⟨
B
,
F
⟩
of some axiom system of set theory
T
in the language of set theory is an end extension of
A
=
⟨
A
,
E
⟩
, in symbols
A
⊆
end
B
, if
A
is a substructure of
B
, and
b
∈
A
whenever
a
∈
A
and
b
F
a
hold, i.e., no new elements are added by
B
to the elements of
A
.
The following is an equivalent definition of end extension:
A
is a substructure of
B
, and
{
b
∈
A
:
b
E
a
}
=
{
b
∈
B
:
b
F
a
}
for all
a
∈
A
.
For example,
⟨
B
,
∈
⟩
is an end extension of
⟨
A
,
∈
⟩
if
A
and
B
are transitive sets, and
A
⊆
B
.