Wiles' proof initially involved reference to functional equivalents of inaccessible cardinals (here, Grothendieck universes). Rather than take this as evidence for the meaningfulness and usefulness of higher set theory (actually, not a much higher level of such a theory, all things considered), it seems as if many mathematicians were/are exercised to show that the proof can (or even "ought to") be reformulated without reference to inaccessible cardinals.
The "version of set theory" I'm working on depends heavily on the question of social constructivism in mathematics.V So I quote here Randall Holmes, from a discussion he was having back in 1998:
If one does not acknowledge that there is real formal structure in the world, then one cannot understand the objective character of mathematics. Formal structure (whether it exists Platonically or inheres in a more Aristotelean manner in objects, independently of us in either case) is what mathematics is about.
Now Fermat's vexatious proposition seems to be a structural question about ℕ. But if this structure is not socially constructed (if the question can be posed without direct/first-order reference to the social will of mathematicians, or agents generally acting in a mathematical capacity), why would it be so "upsetting" for Wiles' proof of that proposition to involve reference to inaccessible cardinals?
VOne of the keystones of the system is an "object" denoted as 𝔼, defined as the well-founded set of all sets-knowable-in-a-well-founded-way. Since this set is not an element of itself, it follows that it is (A) a specific set in the ascending hierarchy but (B) we cannot identify which set it is by the means of our ascent. Our awareness of 𝔼 is like being blind and facing the horizon at night, our nonfunctioning eyes set upon the point of the horizon where the sun will first rise in the morning. At any rate, 𝔼 is meant to be used in place of V (or proper classes more generally) to implement Zermelo's talk of V being an "unfinished totality." So not only is |𝔼| a specific large cardinal at any given time, it changes over time (or: which such cardinal it is, changes), as mathematical knowledge socially develops.