Standard techniques (no pun intended) can be used to show that countable nonstandard models of Peano Arithmetic are order isomorphic to $\mathbb{N} + \mathbb{Z} \cdot \mathbb{Q}$. Once we have used the compactness theorem to verify that nonstandard models of PA exist, we can appeal to the Löwenheim–Skolem theorem to ensure that we have countable ones. We can then verify that these models form unbounded dense linear orders of $\mathbb{Z}$ blocks beyond the standard portion and use a back and forth argument to confirm that the nonstandard parts must always be order isomorphic.


My questions concern the structure of uncountable nonstandard models of PA. The Löwenheim–Skolem theorem can be used to show that we have nonstandard models of arbitrarily large cardinality so:


(1) Is there an example of a set $M$ of the form $\mathbb{N} + \mathbb{Z} \cdot D$ where $D$ is a dense linear order such that $M$ is *not* a model of PA?


(2) Can we find arbitrarily large $\kappa$ and nonstandard models of PA of size $\kappa$ such that there will be a **an unbounded** well-orderable subset of size $\kappa$ respecting the ordering relation of the nonstandard model (i.e. an order isomorphism between an ordinal and a subset of the model)?


(3) If the answer to (2) is no, what types of related results can we get?



Uncountable nonstandard models of PA
Question 2 is an immediate consequence of the compactness theorem. Let $\kappa$ be fixed, and for each $\alpha < \kappa$ add a constant symbol $c\_\alpha$ to the language. Add axioms of the form $c\_\alpha < c\_\beta$ for every $\alpha < \beta < \kappa$. The new theory $T$ is finitely satisfiable (every finite fragment is interpretable in the standard model of PA). So $T$ it is satisfiable, and the map $\alpha \mapsto c\_\alpha$ is the desired embedding. 


For question 1, it is known that $D$ cannot have the order type of the reals. There is a nice survey by Bovykin and Kaye [1] that includes this fact and has a lot more information about order types of models of PA. In the comments below, Andrey Bovykin suggests downloading his thesis from his homepage [2]; the paper is a summary of parts of the thesis. 


1: Bovykin, Andrey; Kaye, Richard. *Order-types of models of Peano arithmetic.* (2002) Logic and algebra, 275--285, Contemp. Math., 302; [MR1928396](https://mathscinet.ams.org/mathscinet-getitem?mr=1928396), [DOI: 10.1090/conm/302/05055](http://dx.doi.org/10.1090/conm/302/05055), <http://www.maths.bris.ac.uk/~maaib/orders.ps>


2: <https://logic.pdmi.ras.ru/~andrey/research.html>


