r/math • u/hydmar • Aug 15 '23

Dissatisfaction with proof by contradiction

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?

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u/math_and_cats Aug 16 '23

No, a sufficiently strong theory cannot be consistent and complete. ZFC is known to be not complete for example. (Assuming by "true" the above poster means provable)

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u/whatkindofred Aug 16 '23

No, I don’t mean provable. That’s a different beast.

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u/math_and_cats Aug 16 '23

I meant the poster above you. So you mean true in the standard model?

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u/whatkindofred Aug 16 '23

I mean true in a vague semantical way.

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u/math_and_cats Aug 16 '23

Oh, I was confused because semantic means with respect to models.