TITLE: "Natural" non-standard models of Peano.
QUESTION [9 upvotes]: The standard model of Peano is particularly natural, being (among other things) the unique model that embeds into any other model of Peano. It's well known that there are many other models of Peano, for example, ones in which there exist non-standard length proofs of the inconsistency of Peano.
However, all methods I've seen of constructing non-standard models never in any sense uniquely specify any non-standard model (but rather use the compactness theorem, though I'm sure there are other ways), and my question is if it is possible to specify a particular non-standard model in some (preferably "natural") way.
This is related to my previous question here, though more broad. (One connection being that if there was a unique model of Peano which embeds in every model except the standard one, that would be a specification of a particular non-standard model, and the poset described in that question can capture that, although I'm sure there is no such model). I haven't studied model theory, just read some interesting things online, so I apologize if this question is ill-posed.

REPLY [7 votes]: This answer is a remark on "all methods I've seen of constructing non-standard models never in any sense uniquely specify any non-standard model". It shows that you can uniquely specify a non-standard model of PA, but it does not answer your question, in the sense that this unique specification is not an order-theoretic characterization among a class of models of PA.
Extend the language $\mathcal{L}$ of Peano arithmetic with Henkin constants to obtain the language $\mathcal{L}_H$. Now, fix an enumeration $f: \mathbb{N} \rightarrow \mathrm{Form}(\mathcal{L}_H)$ of all the formulae of $\mathcal{L}_H$. There are even computable such enumerations, and computable functions can of course be specified uniquely, so fixing such an enumeration does not sneak in any arbitrary choice.
The idea is to extend the theory $\mathrm{PA} + \neg Con(\mathrm{PA})$ to a complete theory $T$ in the language $\mathcal{L}_H$ by iterating over $f$. Set $T_0 = \mathrm{PA} + \neg Con(\mathrm{PA})$. Set $T_{n+1} = T_n + f(n)$ if $T_n + f(n)$ is consistent, and set $T_{n+1} = T_n$ otherwise. Finally, let $T= \bigcup_{n\in\mathbb{N}} T_n$. Observe that $T$ is consistent, complete, and uniquely specified as a function of $f$.
Now, you can obtain a model $M$ of $T$, whose elements are $\equiv$-equivalence classes of terms of the language $\mathcal{L}_H$, where $s \equiv t$ precisely if $T \vdash s = t$. This is essentially Henkin's model construction (see e.g. Lemma 11.4 in Andrew Marks' notes here), and it defines $M$ uniquely in terms of $T$.
So you can start with your favorite, uniquely specified enumeration $f$. The choice of $f$ pins down $T$ uniquely, and in turn $T$ pins down $M$. And $M$ is a non-standard model, since $M \models \neg Con(\mathrm{PA})$. So $M$ is a uniquely specified non-standard model of Peano arithmetic.