When does a proof **require** the use of a specific theorem?

Question

As the title states, I am looking for some explanation of when a result inherently depends on a specific theorem. Another formulation of the question could be

Given two arbitrary proofs, $X_1$ and $X_2$, of a theorem, which facts must always arise in both $X_1$ and $X_2$?

Specifically, I am looking for nontrivial facts. Obviously, statements such as axioms and small formulations based on the axioms are required, but this is not interesting.

An example would be using Bezout's lemma to prove that $(\mathbb{Z}/p\mathbb{Z})^*$ is a group (specifically, showing that inverses exist).

Answer 1

It's pretty unclear what it would even mean to ask when you can, in the course of proving a true statement, avoid the use of a true theorem, since it doesn't obviously make sense to ask "what if the theorem were false?" You can try to avoid citing the theorem, but maybe in the course of the proof you secretly end up reproving it anyway; it's hard to draw a principled distinction here.

However, you can get around this by trying to find a natural generalization of the true statement in which the true theorem you're applying becomes a hypothesis, and ask what happens if the hypothesis is false. For example: typically the way we prove unique factorization for $\mathbb{Z}$ is using the Euclidean algorithm. You might want to know if this is necessary. It's unclear what it would mean to ask this question of $\mathbb{Z}$, but you can generalize by asking this question of other rings, which leads to the following question:

Is every unique factorization domain (UFD) a Euclidean domain (ED)?

The answer, very interestingly, is no. For example, you can show that the polynomial ring $k[x, y]$ in two variables over a field is a UFD, but it can't be an ED because it's not a PID. A more interesting example is that the ring of integers $\mathbb{Z} \left[ \frac{1 + \sqrt{-19} }{2} \right]$ is a UFD, even a PID, but still not an ED; see for example this math.SE question.

We can even make some progress on the corresponding question about $\mathbb{Z}$: it's a general fact that every PID is a UFD, so if you can prove that $\mathbb{Z}$ is a PID without using the Euclidean algorithm, perhaps by proving some more general fact about a larger class of rings that includes non-EDs, then you can really start to claim that you've avoided the Euclidean algorithm. It might be possible to do this using some more general technique for computing the class numbers of number rings, although one has to be careful to avoid circularity here in case any of those results rely on the assumption that $\mathbb{Z}$ is a UFD already...

Answer 2

There is actually mathematics that studies this! The two main approaches are called Reverse Mathematics and Computable Mathematics. To quote the introduction to the article Slicing the Truth

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But suppose we were to take seriously the task of proving that, say, the Four Color Theorem implies that there are infinitely many primes. What are the chances that any of us could come up with a proof that “really uses” the Four Color Theorem? The exercise may seem as pointless as it is difficult, but of course mathematicians do set and perform tasks of this kind on a regular basis. “Use the Bolzano-Weierstrass Theorem to show that if f : [0, 1] → R is continuous, then f is uniformly continuous.” is a typical homework problem in analysis, and the question “Can Chaitin’s information-theoretic version of Godel’s First Incompleteness Theorem be used to prove Godel’s Second Incompleteness Theorem?” led to a lovely recent paper by Kritchman and Raz

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In this article, we will discuss two closely related approaches to making mathematically precise sense of this idea of establishing implications and nonimplications between provably true principles: computable mathematics and reverse mathematics.

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Roughly speaking, reverse mathematics picks an suitably small axiomatic basis to work over, adds a notable theorem, and then sees what must be true in order to get to that theorem from those axioms. Computable mathematics studies fields analogous to those in regular mathematics, but considers "computable" objects - objects that can be suitably recognized by a Turing Machine - and studies the theory that they give rise to, as well as how far from being computable the normal theory is.