The order-type of models of Peano Arithmetic Let $M$ be a non-standard model of Peano Arithmetic. It is well known, that $(M,<)$ has order type $\mathbb{N}+\xi\mathbb{Z}$, where $\xi$ is a dense, linear order without endpoints.
However not every dense, linear order without endpoints is realized in this way, e.g. there is no model of $\mathrm{PA}$ with order type $\mathbb{N}+\mathbb{R}\mathbb{Z}$.
Are there any general results on what order types $\xi$ can or cannot be realized?
I'm particularly interested in the following: Given an ordinal $\alpha>0$, is there a model $M$ of $\mathrm{PA}$ with order type $\mathbb{N}+\xi_\alpha\mathbb{Z}$, where $\xi_\alpha$ is an $\eta_\alpha$-ordering?
 A: I don't know enough about models of PA to tell you which order types are realized (or whether there's a satisfying answer to this general question). But your specific question about $\eta_\alpha$-orderings is easy to answer, just using basic tools of model theory. 
Any first-order theory admits $\kappa$-saturated models for any cardinal $\kappa$. So let $\alpha$ be an ordinal, and let $M$ be an $\aleph_\alpha$-saturated model of PA. If the order type of $M$ is $\mathbb{N} + \xi\mathbb{Z}$, I claim that $\xi$ is an $\eta_\alpha$-order. 
Let $X,Y\subseteq \xi$ be subsets of size $<\aleph_\alpha$ such that $x<y$ for all $x\in X$ and $y\in Y$. For each $x\in X$, let $a_x\in M$ be an element in the copy of $\mathbb{Z}$ indexed by $x$, and similarly for $(b_y)_{y\in Y}$. Then the partial type $p(z) = \{a_x + n < z \mid x\in X,n\in\mathbb{N}\}\cup \{z < b_y-n\mid y\in Y,n\in\mathbb{N}\}$ is finitely satisfiable and mentions fewer than $\aleph_\alpha$-many parameters, so it's realized by an element $c\in M$. The index of the copy of $\mathbb{Z}$ containing $c$ is strictly greater than all the elements in $X$ and strictly less than all the elements in $Y$.
Edit: Since there are many $\eta_\alpha$-orders up to isomorphism, it's  possible to read your question a different way: Given a particular $\eta_\alpha$-order $\xi_\alpha$, is there a model of PA with order type $\mathbb{N} + \xi_\alpha\mathbb{Z}$? I don't know anything about the answer to this question.
