Can we predict the theorem we need when making a proof?

Question

Sometimes when I’m proving a mathematical problem, I might find out that i still need some hints(some important theorem, hypothesis,etc). Is there any possibility to know(to predict) what we need in the proof? Can the model theory solve this problem? Or something about Gödel’s completeness theorem?

Answer 1

There's sort of two ends to this question, as I see it. First, the technical:

"What do I need to prove this?" is really asking "Of the axioms I have, which ones are necessary for the result I want?" Remember, if it's some important theorem you need or something else that isn't an axiom, you could in principle just prove it yourself and then use it, so you don't technically "need" it. The question of which axioms are necessary to prove a certain result is actually a major field of study, called reverse mathematics - you may want to look into it, it's very interesting! But no, we can't generally know which axioms are necessary for a result, and it's for exactly the reason you suggest - Godel's Incompleteness. The easy example is to consider the collection of axioms $PA + 1 = 0$ (that is, Peano arithmetic together with the false statement $1 = 0$) and say we wanted to prove $1 = 0$. This is a consequence of these axioms, but to know whether we need $1 = 0$ to do this proof we would need to know whether $PA$ is consistent. (This argument is a bit loose, but gives the right idea.)

Next, the practical: Of course, in practice, we do care whether an important theorem is necessary, because we can't really expect to prove them on our own. And no, there's no way to predict whether you'll need it - but as you do more and more proofs in a certain field, you start to get a feel for it. If you talk to mathematicians about a problem in their field, you'll probably hear them say something like "oh, yeah, that's just Fermat's Little Theorem" or "it's just a finite-injury construction, right?" They can predict the technique or the central theorem just because they know the field well enough to know what the "usual" tools are. And, while they aren't always right, they're usually okay at it.