[throwawayjava]

It's OK not to be convinced by an argument, even a mathematical proof, and especially an informal proof.

But Cantor's argument is correct.

If you don't find a mathematical proof convincing even though all the trained mathematicians seem to believe the proof is correct, here's my advice. You should FIRST convert the proof into a sequence of valid deductions in a fixed logic. (If you cannot do this, you don't understand the proof/theorem; perhaps (re-)take a few mathematics courses.) If you can find a mistake in the completely formal proof, then you can convert that mistake into an informal explanation. If your disagreement boils down to a disagreement with a axiom in the formal system even though most mathematicians accept it as an adequate foundations, realize you're getting close to philosophy.

This advice is meant for people at the first phase in Terry Tao's hierarchy of mathematical maturity. So not great advice if you're a genius or a trained mathematician (read: people call you doctor).

> There is only one infinity. It means "repeat". It is simply the interplay of finite state with process. You can think of it as an "infinite loop" in programming. To say that one infinity is smaller than another is to deny that the smaller is infinite. Infinite means without bound.

But "modeling non-terminating loops in a computer program" is NOT the motivation for real numbers, so this foundational criticism of completing the rationals makes absolutely no sense.

[mmarx]

> But "modeling non-terminating loops in a computer program" is NOT the motivation for real numbers

Indeed Turing himself proved that there are non-computable real numbers.