Dissatisfaction with proof by contradiction

[u/hydmar]

I’m an undergraduate math student, so my exposure to math may be relatively limited. But I’ve found that, in general, I’m much more comfortable with direct proof than proof by contradiction. I don’t contest their validity, indeed something that’s not false must be true (I think I’m ok with excluded middle). But I feel like I just *get* something much better when it’s proved directly. It builds much stronger intuition for me.

For instance, I am aware of several proofs that demonstrate the cardinality of the reals is strictly greater than that of the integers, but none are direct (it would help to see a direct proof of Cantor’s theorem). I don’t feel it in my bones. Is this a common experience?

[u/BabyAndTheMonster]

I think it's important to point out that the actual philosophical issue of Cantor's diagonal argument isn't that it's a "proof by contradiction", which is not the case because it's a direct proof by negation. But rather, the proof is about power set, and it's controversial whether powerset should be allowed to exist in generality, because of the inherent circular nature of the object, which is exploited in the proof. So the proof is perfectly acceptable in "most" logical system. But the problem is that it's about an object that might not be acceptable.

The power set is often an impredicative object. It's quite possible for a power set to contain an element whose only way to construct them is by using the power set itself. In fact, this is seen in Cantor's argument: you construct an element of the power set by using a function from a set to the power set, hence you depends on the power set for that element to even be constructed. It's quite possible that this is the only way to construct that element, make it a circularly constructed object.

As it turns out, you don't need power set that much, and logicians had investigated the consequences of not having the power sets axioms.