Colin McLarty, ``What does it take to prove Fermat's last theorem?
Grothendieck and the logic of number theory,''
Bulletin of Symbolic Logic 16(3), 359-377 (2010).
This is an amusing overview of some of the logical aspects of FLT.
To understand the context, however, we need to go over some concepts.
The theory of natural numbers using the Peano Axioms is usually called PA. It is devoted to describing the usual counting numbers (along with 0), 0,1,2,3,4,5,....
The next level up (well, for our purposes) is called Zormelo-Fraenkl set theory, ZFC. This theory talks about sets. But one of those sets is the set of natural numbers and everything that PA can prove about natural numbers can also be proved in ZFC. But, and this is important, there are *more* things that ZFC can prove about natural numbers than PA can. This is part of what Godel proved: there are true (provable in ZFC) things about natural numbers that cannot be proved in PA.
Another level up from ZFC involves looking at collections that are 'too big' to be sets. Such things are called proper classes and two very common axioms systems for them are NGB and MK.
Finally (again, for our purposes), there is the concept of a 'universe'. In a sense, this is a stronger assumption that MK and is denoted by ZFC+U.
Now, the vast majority of mathematics uses ZFC for its set of assumptions. This is considered 'standard set theory'. Any math going beyond it should be explicitly detailed in a work in math.
Now, Fermat's Last Theorem (FLT), is a statement in PA. But, it turns out that Wile's proof uses ZFC+U. The question in the McLarty paper is whether the proof can be done in ZFC or even PA.
The basic point of the paper is that *for results stated in PA*, a proof in ZFC+U can be 'reduced' to a proof in ZFC. And it seems possible that the proof of FLT can be reduced all the way down to PA.
Now, needless to say, the OP does not do anything substantive along these lines. But the basic question of how 'deep' FLT really is remains an interesting question.