User blog comment:P進大好きbot/Whether Rayo's number is well-defined or not/@comment-73.15.199.22-20180531171758/@comment-35470197-20180531215514

Actually, there is no description of axioms in the original definition.

However, the notion of natural numbers does not make sense if you do not assume axioms, because first/second order logic without axioms itself does not contain arithmetic.

Indeed, if you could define the notion of natural number without axioms, then every formal theory described in first/second order logic would have the notion of natural number in appropriate way where 0 and 1 are defined and 0 \neq 1 is provable. But, there are many formal theory admitting explicit finite models such as group theory, ring theory, category theory, and so on. It is a contradiction.

So if you consider that Rayo's number has nothing to do with axioms, then it is not well-defined at all. So it is non-sense to ask whether it is well-defined or not any more. That is why I assumed axioms here.