Non-standard integers?

Question

Edit: Realized almost immediately that this was a stupid question - see comment below. Was about to delete it when an answer appeared that seems like saving...

Context: I just saw a presentation of a "senior homors thesis" consisting of more or less empty speculation on applications of non-standard analysis to number theory.

Actually I missed the talk, heard about it afterwards. The advisor didn't have any actual applications in mind, was just speculating on how it might be that things like $\Bbb Z_p$ for an infinite prime $p$ could be interesting. Probably just as well I wasn't there, because I might have asked this:

Q: Before we can talk about infinite primes we need to know what an infinite integer is. Is there a "standard" definition of the non-standard integers as a subset of the non-standard reals?

My non-standard reals tend to come from the Compactness Theorem in logic. So I'd know what a non-standard integer was if I had a first-order definition of $\Bbb Z$ as a subset of $\Bbb R$; I can't imagine how that would go. Is $\Bbb Z$ definable in the first-order theory of $\Bbb R$?

(Yes of course if we want to do non-standard number theory we could just as well start by applying the compactness theorem to the theory of $\Bbb Z$; that doesn't quite answer the question...)

Answer 1

An application of $Z_p$ for infinite $p$ was given by Alexei Belov in the context of the Jacobian conjecture in algebraic geometry. See https://arxiv.org/abs/math/0512171 and later papers.

Answer 2

The integers, as a subset of $\mathbb{R}$, cannot be defined in the first-order language of ordered fields: i.e. using the constants, (partial) operations, and relation $0,1,+,-,\cdot,{}^{-1},\leq$.

This fact arises from the theory of real closed fields, which has the notable feature of being complete.

Gödel's incompleteness theorem therefore proves that one cannot identify the integers, because that would let you reproduce Peano arithmetic. However, if I recall correctly, more can be said:

• the only subsets that can be so defined are finite unions of intervals
• of those, the only ones that can be defined without parameters are those where the intervals have algebraic (or infinite) endpoints

---

However, the (IMO) more generally useful formulation of nonstandard analysis doesn't take a theory of real numbers as the starting point — it takes a theory of sets.

In particular, one uses their favorite means to obtain an elementary extension of a superstructure which knows about all of the sets one would use in the course of real analysis.

In the way I know to carry out construction, $\mathbb{Z}$ is a constant symbol of this theory, so its trivial to identify it.