Connection between number theory and the Von Neumann construction of naturals

Question

There are many unsolved conjectures and hypothesis in number theory. For example, the twin primes conjecture, Goldbach's conjecture, the Riemann hypothesis, infinitude of Mersenne's primes, and many many more.

A widespread convention of "rigorous construction of the number system" is von Neumann's finite ordinals, constructed with ZF axioms, i.e. well-ordered transitive sets ordered by the $\in$ relation. It can be shown that these finite ordinals satisfies Peano's arithmetic axioms.

As an electric engineer who designs and constructs a building knows all about the power sockets in the buildings, I would expect that the construction of the natural numbers should imply many things about the behaviors of those numbers, in particular prime numbers.

So my question is, is there a relation? Are there any clues in the way we construct the numbers that can help us reveal more about numbers? Are there any set-theoretic characteristics of prime numbers? Unlike geometry and complex analysis, why doesn't set theory provide us with heavy tools to deal with the complicated world of numbers?

Answer 1

We do indeed have heavy tools to study numbers, which have been constructed with the aid of set theory.

But first, the primary role of set theory is not so much to provide us with "heavy tools", but instead to provide us with a "solid foundation". To be more precise, set theory provides us with a single foundation upon which we may build all of the heavy tools: natural numbers, integers, real numbers, geometry, complex analysis, etc.

The point here is not so much the specific details of any particular set theoretic construction. When studying integers, mathematicians do not work with a specific construction such as Von Neumann's ordinals. Nor, when studying real numbers, do mathematicians work with Dedekind cuts. Instead, once integers or real numbers are solidly founded on an axiomatic basis (itself built using set theory, Von Neumann ordinals, Dedekind cuts, or whatever works), mathematicians have the confidence to proceed with new constructions needed to study whatever they want to study. Often set theory gives them the tools they need to carry out those constructions.

With the set theoretic foundation in place, mathematicians put the tools used to study numbers on a solid foundation as well. And there are indeed many such tools which may have been viewed suspiciously before set theory but are now solidly founded.

For example, Kummer's theory of ideal numbers was solidly established on a set theoretic basis by Dedekind, and the unique factorization theorem for ideals in number fields, which says that any ideal may be factored uniquely into prime ideals, is regarded as a key tool in generalizing (and studying) the unique factorization of integers into prime integers.