Lemma 2.2.9 (Godel’s Lemma). Assuming that:

$M ⊨ PA$
$n \in ℕ$
$x_{0}, \dots, x_{n - 1} \in M$

Then there is

u \in M

such that

M ⊨ {(u)}_{i} = x_{i}

for all

i < n

.