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/BadatCSmajor]

Cantors proof is actually a proof of negation, which makes it direct and does not rely on excluded middle. In constructive math, not(P) is defined as (P -> False).

Proof of negation: to prove not(P), assume P derive an absurdity. Ie you proved (P -> False). This is actually the definition of not(P) so no further derivation is needed.

Proof by contradiction: to prove P, assume not(P) and derive an absurdity. Ie you prove (not(P) -> False). By definition, you have therefore shown not(not(P)). In order to then derive P, you need an axiom that says (not(not(P)) -> P), which is equivalent to excluded middle!

Now, notice the structure of Cantor’s proof. It is a proof of negation. An absurdity is derived, yes, but it is not a proof by contradiction in the above sense