Let
A
\mathcal {A}
be a structure and let
U
U
be a subset of
|
A
|
|\mathcal {A}|
. We say
U
U
is extendible if whenever
B
\mathcal {B}
is an elementary extension of
A
\mathcal {A}
, there is a
V
⊆
|
B
|
V \subseteq |\mathcal {B}|
such that
(
A
,
U
)
≺
(
B
,
V
)
(\mathcal {A},U) \prec (\mathcal {B},V)
. Our main results are: If
M
\mathcal {M}
is a countable model of Peano arithmetic and
U
U
is a subset of
|
M
|
|\mathcal {M}|
, then
U
U
is extendible iff
U
U
is parametrically definable in
M
\mathcal {M}
. Also, the cofinally extendible subsets of
|
M
|
|\mathcal {M}|
are exactly the inductive subsets of
|
M
|
|\mathcal {M}|
. The end extendible subsets of
|
M
|
|\mathcal {M}|
are not completely characterized, but we show that if
N
\mathcal {N}
is a model of Peano arithmetic of arbitrary cardinality and
U
U
is any bounded subset of
N
\mathcal {N}
, then
U
U
is end extendible.