Lemma 2.2.19
(Overspill)
.
Assuming that:
M
a non-standard model of
PA
φ
(
x
)
an
L
PA
-formula
M
⊨
φ
(
n
)
for all standard natural numbers
n
Then
there is a nonstandard natural number
e
such that
M
⊨
φ
(
e
)
.