Let $M$ be a nonstandard model of $PA$. 

> Q1. Is there any way to get a submodel $N\subset M$ such that $N\models Th_{\Pi_2}(PA)$, but $N\not\models PA$?

> Q2. Especially, what combinatorial principle can be used to construct a model $N\models Th_{\Pi_2}(PA)$ such that $N\not\models PA$? Or $Th_{\Pi_2}(PA)\vdash PA$?