Natural numbers are "natural" natural numbers. That is, I would argue, that the Peano definition is the most simple, straightforward and obvious (not that I wou

Natural numbers are "natural" natural numbers. That is, I would argue, that the Peano definition is the most simple, straightforward and obvious (not that I would have come up with Peano's axioms) and therefore "natural" way to define something like whole numbers. And interestingly it is the prototypical way to define an (countably) infinite set, which is the actual cause of the trouble. If you limit everything to finite sets of stuff, all the Goedel evilness goes away. But then, math would be boring. If you admit non-finite sets, you will have countably infinite sets, and those will always be isomorphic to natural numbers. I'd bet good money that if there was an alien civilization, there would be an alien named Peano.