So I recently just finished a course on Peano arithmetics and it's non-standard models. I am very much intrigue by this topic. Hence I am rather curious what are some research areas surrounding PA right now ?
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It's a very rich subject, but it can be hard to navigate. Rather than try to describe the subject, let me suggest a few specific sources; even if they're not quite what you're looking for, I think they'll help you get started (and their introductions and bibliographies will help you find what you do want).
Kaye's Models of Peano arithmetic and Kossak/Schmerl's The structure of models of Peano arithmetic are the most obvious ones to mention. These are the only two textbooks on PA that I know of. I've listed them in chronological order, but they're both nicely self-contained; personally I prefer the latter.
Subtheories of PA (e.g. I$\Sigma_n$) are much more easy to navigate, in my opinion. Here the book I recommend is Hajek/Pudlak's The metamathematics of first-order logic. The study of weak subtheories of PA and the study of PA itself have very different flavors, but they're both worth diving into.
That covers the standard texts as I see them; let me mention a couple more sources which are more difficult but which you might still find valuable.
You specifically mentioned models of PA, but another very important topic here is proof theory. Godel's theorem says that PA cannot prove its own consistency; however, we can still ask whether we can find a "reasonably simple" system which can prove Con(PA), and in particular hope for some way of measuring the difficulty of proving Con(PA). The first step in this direction was taken by Gentzen, who showed roughly that the consistency of PA can be proved by a very weak base theory (PRA; much weaker than PA) together with an additional assumption: roughly that the ordinal $\epsilon_0$, which you can figuratively think of as $$\omega^{\omega^{\omega^{...}}},$$ is in fact well-ordered. That is, a certain amount of transfinite induction captures the difficulty of proving Con(PA). Moreover, Gentzen showed that no smaller ordinal would do: $\epsilon_0$ is exactly what you need. This was the first result in what became known as ordinal analysis, for which I recommend Rathjen's survey paper (it quickly becomes hard going, but it should still give you enough of a taste of the subject for you to know if it's something you're interested in).
Finally, getting back to a model-theoretic flavor, I think some of Bovykin's papers may be interesting to you, e.g. this one or this one.
Noah Schweber
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