Proposition 2.2.21. Assuming that:

$C (u, x)$ be a $Δ_{0}$ -formula
$M$ a non-standard model of $PA$

Then given any

\tilde{b} \in M

, there is

c \in M

such that, for any

n \in ℕ

:

M ⊨ \exists k < \tilde{b} . C (k, n) \leftrightarrow \exists y . (π_{\underset{̲}{n}} \times y) = c .