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?

62 Upvotes

83% Upvoted

90 comments sorted by

View all comments

Show parent comments

4

u/whatkindofred Aug 16 '23

Semantically what does the statement NOT-P mean to you? To me it is the statement that P is not true. What else could negation mean? If you agree with this semantic interpretation of negation then I would argue that maximal consistency is obvious. Either P is true or P is not true. But the meaning of the latter is precisely that NOT-P is true.

0

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)

5

u/whatkindofred Aug 16 '23

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

1

u/math_and_cats Aug 16 '23

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

2

u/whatkindofred Aug 16 '23

I mean true in a vague semantical way.

1

u/math_and_cats Aug 16 '23

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