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?