Applications of Non-Standard Analysis to Number Theory and Topology?

Question

S.E friends,

I am wondering what might be some useful applications of the non-standard analysis to the number theory and topology? I am very interested in the set-theoretic topology and prime distributions, and I recently came across the field of non-standard analysis...The use of infinitesimal is interesting but I am not sure how can it be applied to my fields of interest.

Do infinitesimals offer different perspective to the analytic number theory and topology?

Answer 1

For an application in topology, consider Tychonoff's theorem. Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. Once the basic machinery of the hyperreal framework is in place, the proof of Tychonoff's theorem is essentially a one-liner. The point is that there is an elegant an convenient characterisation of compactness as follows. A space $A$ is compact if and only if every point of its natural extension $A^\ast$ is nearstandard. Here being nearstandard for $x$ means that that there exists an infinitely close point to $x$ which belongs to $A$.

Thus, $\mathbb{R}$ is not compact because its natural extension contains infinite numbers which are therefore not infinitely close to any real number.

The interval $(0,1)$ is not compact because $(0,1)^\ast$ contains infinitesimals which are not infinitely close to any real strictly between $0$ and $1$.

Answer 2

In number theory, I've seen occasional good use made of the infinite numbers. If $P$ is an infinite prime number, then the internal finite field of $P$ elements is, externally, a field of characteristic zero.

I've used facts about finite fields to construct fields of characteristic zero with nice properties, and used facts about characteristic zero fields to prove things about sufficiently large finite fields.

The first time I've seen this method is in regards to the Ax-Kochen theorem: if Wikipedia's account is accurate, the main idea of the proof can be rephrased as saying that the field $\mathbb{Q}_P$ of $P$-adic numbers and the field $\mathbb{F}_P((t))$ of formal Laurent series over the finite field of $P$ elements happen to be, when viewed externally, elementary equivalent as valued fields.