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Since I don't have technical knowledge, I can't express my question perfectly. For example, on the basis of Fermat's Last Theorem, let's ask a this question:

Solve $a^n + b^n = c^n$ for any integer value of $n$ greater than $2$, where $a,b,c\in\mathbb Z^+.$

Then, our answer is: To find such value of $n$ is impossible. And we can prove this impossibility. I want to ask, using which set axioms we can prove that, this is impossible? Because, as far as I know, a proof must necessarily contain certain axioms of set theory otherwise. Or we do not need to specify it? When we prove any object, how can we understand which fundamental axioms of set theory we use? For example, proof of fermat's last theorem contains which set theory axioms? Generally, which is the most powerful set theory we use? (Set theory taught to us in schools, or containing all calculus topics)

Conifold
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    Frankly, it probably uses all of them implicitly, since even the level of maths you learn in school depends on all of the axioms. Trying to do it in terms of set theory would make the proof ridiculously long. – Rushabh Mehta Jan 13 '20 at 02:42
  • Usually, when you use axioms in a proof you need them all, ex- or implicitly. In this operational sense they are all equal. If you add a meaningless axiom such as "in addition, ice cream taste good" that would constitute a less powerful axiom in your system. – Mikael Jensen Jan 13 '20 at 03:40
  • Likely not axiom of choice. – Michael Jan 13 '20 at 04:18
  • Followup: I assumed the Wiles proof of Fermat's theorem did not use axiom of choice (which motivated my above comment) but from a quick google search I find this link which suggests it may or may not use axiom of choice (I cannot make heads or tails of the link as it seems to be an argument against asking the question, and links to deleted questions, rather than answering the question!) https://meta.mathoverflow.net/questions/1336/why-was-this-question-about-axiom-of-choice-and-the-wiless-proof-of-flt-deleted – Michael Jan 13 '20 at 04:28
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    @Michael Wiles’ proof may. However assuming his proof is correct (or it is provable by other means), there is definitely a proof that doesn’t use AC. – spaceisdarkgreen Jan 13 '20 at 04:41
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    @spaceisdarkgreen : On your comment "there is definitely a proof [of Fermat's last theorem] that doesn't use AC": I assume you would have provided a truly marvelous reference to this, but the margin of your comment field is too narrow to contain it. =) – Michael Jan 13 '20 at 04:46
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    @Michael haha. Actually looking at the link you posted it is talked about there. Schoenfield’s absoluteness theorem implies sufficiently simple number-theoretical statements never require AC to prove. – spaceisdarkgreen Jan 13 '20 at 04:50
  • Grothendieck's Universes & Wiles' proof of Fermat's last theorem. Check out this. https://www.jstor.org/stable/20749620?seq=1 – SL_MathGuy Jan 13 '20 at 06:17
  • @Michael (Minor correction: it's "Shoenfield" and more importantly, FLT is a first order statement, so using his theorem (which establishes this for some second order statements as well) is major overkill here, which is not to say it doesn't work.) – spaceisdarkgreen Jan 13 '20 at 06:34

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Many experts suspect that Wiles's proof of FLT can be converted into a proof in Peano arithmetic which is much weaker than set theory, see Are there non-standard counterexamples to the Fermat Last Theorem? and What is known about the relationship between Fermat's last theorem and Peano Arithmetic?

If this is possible then we only need axioms of set theory sufficient to derive Peano arithmetic. This would mean that axioms of choice, infinity and replacement are not needed, they are used to deal with infinite sets. What is left is a theory of finite sets, which is more or less equivalent to arithmetic, see Systems the Peano axioms can be derived in. But so far whether it can be done or not is an open question. As Colin McLarty writes in What Does It Take to Prove Fermat's Last Theorem:

"Angus Macintyre lays out a program to express the Modularity Thesis (MT) central to Wiles [1995] as a statement of arithmetic and argues that it is provable in PA. This program could lead to a PA proof of MT, and possibly one of FLT without using MT. It calls for substantial new work in arithmetic. While closely based on Wiles [1995] it is no routine adaptation."

Generally speaking, tracking which axioms are actually used in a proof is a tedious but routine exercise. It is not very informative though. Most proofs are using results whose use can be avoided to simplify or shorten proofs. Wiles's proof uses the aformentioned modularity thesis, "universes", axiom of choice constructions, and other very strong tools that are not essential. As is it uses all ZFC axioms. Calculus, as is, uses them all as well (possibly excepting replacement), but most of it can be done without the axiom of choice.

On the other hand, figuring out what is minimally needed to prove a complex theorem is a non-trivial question in its own right, and is related to the so-called independence results, a field in mathematical logic.

Conifold
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