That's a good question, actually. I'm Not OP, I'm sure OP "understands" them as much as anyone does. But considering that the most common definition is an incomplete extrinsic definition by example, N = {1, 2, 3 ...}, I'd argue that in principle no complete extrinsic definition can be given :) That's more than a solipsism, because any number has inherent properties that make it different from all the others, even if these properties might be equal up to isomorphism with "n'th successor to zero", because then the system of isomorphisms is in question, begging the question ...

Whereas, if you know the intrinsic definition by the axioms, you know the definition, not the numbers. Big difference that is.

I'm keen on a distinction between numbers and forumals. If you take binary numbers and succ(), you have 0, 1 and an infinity of formulas. In my book that's only two natural numbers. We commonly change base to represent eg. 16 as 0x10 - or 1000 as 1k, requiring additional figures. Figure, number - potato, potato.