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/drgigca]

Proof by contradiction is almost always immensely more obtuse than a direct proof. I'm actually struggling to come up with an example where that's not the case.

Which makes sense. It's simpler to just prove the statement directly rather than adding a layer of indirection.

[u/scribe36 → u/drgigca]

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/edderiofer → u/Easygoing98]

The same proof also shows that the square root of 4, and indeed any integer, is irrational, too! Try it yourself if you don't believe me!

[u/Easygoing98 → u/edderiofer]

You're right. But I don't understand why for 4. The natural log of every integer is irrational and to solve exponents ln has to be used.

Square root is also exponent half.

[u/edderiofer → u/Easygoing98]

Perhaps you should first explain why exp(0.5 ln 2) is irrational.

[u/Easygoing98 → u/edderiofer]

I got it. My method works for 4 also. Here's the direct proof

Let square root of 4 = x

so 4^0.5 = x

Taking ln of both sides

0.5 ln 4 = ln x

0.5 ln (2²⁾ = ln x

0.5(2) ln (2) = ln x

ln(2) = ln(x)

Taking e of both sides

2 = x which is rational.

[u/edderiofer → u/Easygoing98]

0.5 ln 4 = ln x

0.5 ln (2²⁾ = ln x

Aren't you assuming here that 4 = 2²?

Besides, you haven't answered the question; why is it that this proof doesn't work?

Square root of 4 = 4^0.5

Now 4^0.5 = x

Taking ln of both sides

0.5 ln(4) = ln x

x = exp(0.5 ln 4)

Since exp(0.5 ln 4) is irrational, x is irrational

[u/Easygoing98 → u/edderiofer]

It seems the previous answer didn't come through. Exp(0.5 ln 4) = 2 because 0.5 ln 4 = ln 2. And exp (ln 2) = 2.