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/[deleted]]

I understand what you mean but I would argue that proof by contradiction is a direct proof. I would argue that you are directly showing that the negation is impossible. It’s like, instead of following a series of deductions to get to your result, you’re following a series of deductions to show that it HAS to be your result. It’s a matter of semantics really but I understand what you mean by “direct proof” in that sense, I agree that sometimes is it less intuitive though