# Connection between number theory and the Von Neumann construction of naturals

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?

• I'm somewhat uncomfortable with these sorts of question, because I feel that they don't belong into the realm of mathematics and only allow for incomplete and wrong answers. Nevertheless, I tried to come up with a satisfactory answer - and failed. Let me leave you with its only part, which I feel adds to your post: The only proof of (the somewhat number theoretical) Goodstein's theorem that I know of, uses set theory. There is a second one, that deals with arithmetic progressions, but I don't recall its name or precise statement. – Stefan Mesken May 19 '16 at 15:06