I know how how natural numbers work, but the axioms in the comment i replied to are not enough to define them.

Not sure what you mean by ‘loops’

There could be a number n such that m=s(n) and n=s(m). This would be precluded by taking the axiom of induction or the trichotomy axiom.

If we only take the latter we can still make a second number line, that runs “parallel” to the “propper number line” like:

n,s(n),s(s(n)),s(s(s(n))),...
0,s(0),s(s(0)),s(s(s(0))),...

there are no natural numbers that are negative

I know, but the given axioms don’t preclude it. Under the peano axioms it’s explicitly spelled out:
0 is not the successor of any natural number