What are some fun nonstandard models to explore?

Question

I juts got introduced to my first nonstandard model of arithmetic, which has elements in its domain that are bigger than any natural number. I really liked it, so I'm interested to hear some more examples of such fascinating models. Thank you.

Answer 1

Peano arithmetic and even more real analysis are the standard application of non standard models. But even more interesting are the application in measure theory and combinatorics.

Google Loeb measure or have a look to Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory for an introduction.

If you understand "non standard models" in a broader sense, any result of (applied) model theory is an application of "non standard models".

Answer 2

Amazing question! The first non-standard model that I recommend viewing is the one of $\mathbf{PA}$, which is also the ultrapower of the standard model of $\mathbf{PA}$. You can delve into nonstandard analysis and prove the properties of the reals, their order, and their completeness through a nonstandard fashion. It also becomes apparent how important Los's theorem is and how important the compactness theorem is which is usually mentioned in textbooks very early on while it is perhaps one of the few ways that can prove consistency.

I'll give a glimpse by showing some consequences of non-standard models of the reals, and the rest you can delve into on your own, and I'll leave some book recommendations at the end.
Given a standard model of the reals, $\mathbb{R}$, then the nonstandard model can be given through an ultrapower with respect to a non-trivial ultrafilter $\mathcal{U} \subseteq \wp(\omega)$. Our ground model is $\mathbb{R}$ which so happens to be our domain also. The nonstandard model will be $\mathbb{R}^{*}$, which domain will be denoted in the same way, and the natural numbers for the ground model (standard model), and the nonstandard one will both be the same notation-wise. We say that $\overline{\mathbb{R}}$ is the set of all reals $r^{*} \in \mathbb{R}^{*}$, with $r^{*} \in [x_1, x_2]$ which belong to the ground model ($x_1, x_2 \in \mathbb{R}$). We say that an $\epsilon$ is an infinitesimal such that it is equal to ${1\over \delta}$, with $\delta \in \mathbb{N}^{*}{\setminus}\mathbb{N}$. In the ground model, $\epsilon$ is smaller than any $1 \over n$ (for $n \in \mathbb{N}$) which means it does not exist in for it but in the nonstandard model it is a positive rational, and the nonstandard counterparts of $x_1$ and $x_2$ ($x_{1}^*$ and $x_{2}^*$) are said to be infinitely close. Meaning that $x_{1}^* - x_{2}^*$ is an infinitesimal. There also exists an equivalence relation on the nonstandard model that could denote the property of being infinitely close. This relation also applies to unique reals in the ground model (standard model) $x_{1}^* \equiv x$, with $x \in \mathbb{R}$. (This is called a nonstandard projection)

Book recommendations: Nonstandard Analysis, Axiomatically by Kanovei, Nonstandard Models of Arithmetic and Set Theory by Kossak R.
Hope this helped. :)