It's easy enough to show that if $\mathbb{N}\_1$ is a non-standard model of the Peano axioms, then there is a canonical embedding $\mathbb{N} \to \mathbb{N}\_1$, and we have a theorem that if $x \in \mathbb{N}\_1$ and $y \in \mathbb{N}$ such that $x < y$, then $x \in \mathbb{N}$.


What if we had two non-standard models $\mathbb{N}\_1 \subseteq \mathbb{N}\_2$, ideally an elementary embedding? Must it be true that if $x \in \mathbb{N}\_2$ and $y \in \mathbb{N}\_1$ such that $x < y$, then $x \in \mathbb{N}\_1$? 


I'm also curious about the analogous questions for models of real analysis or of ZFC; e.g. comparing the set of integers in two nonstandard models of analysis.



Is a model of arithmetic contained in a model of arithmetic an initial segment?
Let me address the first question.


We can have models of $\mathsf{PA}$, $M\subsetneq N$ with $M$ cofinal in $N$. In fact, $M$ and $N$ do not even need to have the same cardinality. However, one can prove from the Davis-Matiyasevich-Putnam-Robinson theorem (on Hilbert's tenth problem) that already the assumption $M\subseteq N$ implies that $M$ is $\Sigma\_0$ elementary in $N$. This and having $M$ cofinal in $N$ suffice to imply that in fact $M$ is an elementary substructure of $N$. 


This is a basic result on models of arithmetic. Kaye's book should have the details. Gaifman improved this by showing that whenever $M$ and $N$ are models of $\mathsf{PA}$ with $M\subseteq N$ then, letting $L$ be the downward closure of $M$ in $N$, that is, $$ L=\{a\in N\mid \exists b\in M\,(N\models a\le b)\}, $$ we have that $L$ is also a model of $\mathsf{PA}$, $M$ is an elementary substructure of $L$, and $N$ is an end-extension of $L$. 


