Comment by im3w1l

I feel like we have some miscommunication. I'm referring to this list of axioms: https://mathworld.wolfram.com/PeanosAxioms.html

And I asked myself what happens if we do away with axiom number 5?

As for the nonstandard integers, I think that's a different thing.

There also is apparently already a concept in math named the supernatural numbers (aka Steinitz numbers) but those were not the ones I meant either.