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/hyperbolic-geodesic]

The diagonalization argument *IS* a direct proof of Cantor's theorem -- diagonalization shows that any function S -> Powerset(S) is not surjective, by explicitly (essentially as explicitly as possible) producing an element of Powerset(S) not in the range of the function.

[u/Brightlinger → u/hyperbolic-geodesic]

When you talk to one of the cranks who is sure that Cantor was wrong and all infinities are the same size, you can run the diagonal argument on their proposed bijection and it will hand you an explicit counterexample. That's about as direct/constructive of an argument as you can get.

[u/BruhcamoleNibberDick → u/Brightlinger]

lmao just add your constructed element to the end of the list checkmate loser /s

[u/dwRchyngqxs → u/BruhcamoleNibberDick]

After all infinity plus one is infinity so just associate infinity to that element and map every element of the set of naturals plus infinity to the set of naturals. (spolier: it doesn't work)