What is the mathematical definition of "standard arithmetic/standard natural numbers"?

Question

As a consequence of Godel's incompleteness theorem, no axiomatic system can be the definition of standard natural numbers because any axiomatic system of arithmetic will always be satisfied by non-standard models too. E.g.-

1. Robinson's arithmetic may be satisfied by models whose elements are neither even nor odd

2. Peano arithmetic may be satisfied by models whose elements do not satisfy Goodstein's theorem

The natural number system that humans intuitively know are either even or odd and they do satisfy Goodstein's theorem

But what exactly are the natural numbers that humans intuitively know? What's their mathematical definition? Since no axiomatic system (even ZFC) can be taken to define them, is there some non-axiomatic definition of them?

Answer 1

Your use of the word "intuitive" means that we're entering philosophical waters.

In ZFC, as you know, one can prove the formal version of the assertion "there is, up to isomorphism, only one model of the second-order Peano axioms". But as you are also aware, there are non-standard models of ZFC (assuming it's consistent), and these can have non-standard omegas.

Now to someone who adopts a certain kind of formalist attitude, there's no more to be said. For an intuitionist like Brouwer, on the other hand, our intuition of the natural numbers is basic, and not something that one can (or should even try) to formalize.

For most Platonists, it's simply assumed (an "article of faith") that the entities of mathematics exist in some "universe of abstractions". $\mathbb{N}$ is then a denizen of this universe. Our axiom systems, like PA and ZFC, capture some of the "facts" about this universe that we somehow know.

If you're asking for a precise mathematical definition of the natural numbers, I would say either "there isn't any", or "omega, working inside the ZFC axioms, or something along those lines".

Consider why you're sure that Goodstein's theorem holds for "the standard natural numbers". Surely it's because you believe in ZFC, in some sense. Or at least some fragment of ZFC. In other words, the power set of $\omega$ is ascribed some sort of reality.

Now consider ZFI, which is ZFC plus "there is an inaccessible cardinal". The consistency of ZFI does not follow from the consistency of ZFC. The statement Con(ZFI) is, however, a statement in PA (but of course not provable in PA). Are you sure it's true? Likewise for even stronger large cardinal axioms, or other extensions of ZFC that cannot be shown to be relatively consistent.

As it happens, John Baez and I carried on a lengthy discussion of these matters. Baez said:

Roughly, my dream is to show that “the” standard model is a much more nebulous notion than many seem to believe.

Along the way we considered the "ultra-finitists", who consider even such a notion as $2^{100}$ as unclear.

You'll find the conversation here. Post 5 and post 6 especially focus on your question.

One last comment: regardless of one's philosophical stance, if someone says "This is true of the standard natural numbers", the purely mathematical content of this claim usually amounts to "this can be proved for $\omega$ in ZFC" (or some fragment of ZFC). At least that's how I would understand their assertion.