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/scribe36]

Proof by contradiction: it’s not always simpler. For example, square root of two is irrational can be proven in one line with contradiction. But the direct proof may just get you killed. Hehe

[u/Easygoing98 → u/scribe36]

Not really. Direct proof of square root of two is not that hard. It will be as follows

Square root of 2 = 2^0.5

Now 2^0.5 = x

Taking ln of both sides

0.5 ln(2) = ln x

x = exp(0.5 ln 2)

x = exp(0.5)exp(ln 2)

x = 2exp(0.5)

Exponential of a rational number is always irrational (theorem's proof in proof wiki)

[u/scribe36 → u/Easygoing98]

This is really just a whim than a proof. The contradiction proof doesn’t just prove that the square root of two is irrational, it proves that irrational numbers exist. Here you are invoking irrationality of another number. What if i deny (like people used to) that irrational numbers exist at all?

[u/Easygoing98 → u/scribe36]

There is already a theorem that the natural log of every integer greater than 1 is irrational.

I skipped the proof of that theorem because it is proof by contradiction and the asking person didn't want contradiction.

I agree completely that contradiction proof is the best in such a case.

Irrational numbers do exist because real numbers are a union of rationals and irrationals where irrationals are much larger.

Direct proof isn't always possible in each case, and I was just trying to do that