We must remember that these constructions retcon the actual human experience with numbers, which is firmly rooted in counting and measurement, that always come after this experiential basis. Formal definitions appear driven by an aesthetic consensus, mostly about minimalism and axiom flavor. Individual human experience with numbers reflects human history and at both scales 'counting and measurement' is more fundamental in the sense that it always comes first, and therefore all constructions derive from it, must maintain consistency with it, and conflicts must be resolved within it, and it is safe to ignore inconsistencies that can't be expressed within it.
Without formal constructions, humans have a nasty habit of proving things that wrong or not even wrong. Our ancestors dealt with this problem in the 19th Century by creating tools to help distinguish truth from balderdash.
Your statement is consistent with my point which I mean to point out the psycho-social origin of such constructions and therefore the limits of their utility. I do not say they are not useful at all.
In those terms, the foundations (of any topic or field) are never fundamental. I think people in this thread are using a different meaning of "fundamental".
