non-isomorphic countable models of $Th(\mathbb{N})$

Question

I'm proving there are exactly $2^\omega$ non-isomorphic countable models of standard natural numbers. I got cardinality of them $\geq 2^\omega$from prime arguments. but I don't get how to prove other direction $\leq 2^\omega$.

Answer 1

Let $I$ be a countably infinite index set. There are no more than $c$ ordered pairs $(f,g)$ of functions $f:I\times I\to I$, $g:I\times I\to I$, representing addition and multiplication. Since any countable model of number theory can be taken, up to isomorphism, to have underlying set $I$, there can be, up to isomorphism, no more than $c$ countable models of number theory.