example of nonstandard model of PA that is not recursively saturated

Question

I know that every nonstandard model of PA realizes any recursive type of given quantifier complexity (say $\Sigma_n$, for some $n$). I suppose there must be recursive types that are not always realized in nonstandard models. Can anyone give same simple and/or interensting example of such types?

Answer 1

Recursively saturated models are "tall", meaning that for every element $a$ of the model, the type $p(v) = \{ v > t(a) \}$ where $t$ ranges over all Skolem terms, is realized. This is a recursive type, since we can recursively enumerate the Skolem terms.

If you let $a$ be any nonstandard element, and let $\mathcal{M} = \textrm{Scl}(a)$, the Skolem closure of $a$ (the smallest model of $\textsf{PA}$ containing the element $a$), then that type $p(v)$ is not realized in $\mathcal{M}$. It also is not realized in any cofinal extension of $\mathcal{M}$.