The idea behind Cantor's diagonalization proof is that you can't find any way to assign "first", "second," "third", etc to the reals. The proof assumes that you can, and that you've already assigned every real number a unique natural number.
It then derives a contradiction by proving that there must be reals that aren't in that ordering, without making any assumptions about the ordering. So any ordering has this problem and none of them work.
The proof starts assuming you can count natural numbers, which you can’t.
It’s impossible to have a known cardinality of an “infinite set”.
The definition of set is “a collection of items”. You cannot “have” an infinite collection of items.
You can have a method/function/formula/algorithm that generates as many items as you can in a certain amount of time, but by definition of infinity, you can never “count” an infinite number of elements.
Hence, you can never know the cardinality of an infinite set (like the natural numbers).
Once you think you can reduce something unknowable, to a symbol, you can prove anything.
Sure, that way lies intuitionism, which is a perfectly respectable (if not particularly popular) school of mathematics.
> Once you think you can reduce something unknowable, to a symbol, you can prove anything.
That doesn't follow. If ZFC is trivially inconsistent then someone would probably have noticed by now and proved P && !P for some P and brought it crashing down. That hasn't happened though, so even if you have strong aesthetic preferences against infinite sets being actually real, it seems like you can treat them as if they are real and produce a productive and not-obviously-inconsistent mathematical system. Most mathematicians don't really care if the naturals are actually infinite or just can be productively treated as if they are.
> That doesn't follow. If ZFC is trivially inconsistent then someone would probably have noticed by now and proved P && !P for some P and brought it crashing down.
Ahh, the classic economic argument, “that’s not a $100 bill on the sidewalk, because if it was, someone else would have picked it up already”.
That’s a great way of accepting everything blindly to justify not questioning things.
It also means you are using popularity as a measure of truth.
It’s fine if mathematicians don’t care if naturals are “actually infinite”, but then what’s the point in reasoning about infinity through math if technically you are not really saying anything, but you’re going in circles around your own definitions instead?
It then derives a contradiction by proving that there must be reals that aren't in that ordering, without making any assumptions about the ordering. So any ordering has this problem and none of them work.