> I don't know of a way to prove equipotence except by creating a 1-1 relationship between the sets in question.
The usual definition of two sets having the same size (cardinality) is that they are in bijection. So ultimately, any proof that |A| = |B| will reduce to a "creating a 1-1 relationship between the sets in question", or at least showing that one exists.
Re:Dedekind's Subtle Knife
by andrewflnr on Tuesday February 18, 2025 @11:43AM
(#43091856)
Ok cool, I didn't remember that was the actual definition.
> I don't know of a way to prove equipotence except by creating a 1-1 relationship between the sets in question.
That relies on first defining it as "there is 1-to-1 mapping". But for sequences of numbers you could instead compare growth rates. Then the natural numbers and the even natural numbers wouldn't be the same size. Which is nice, because the latter sequence is a proper subset of the former. The former grow twice as fast, so you could say they are twice as large. But it only works for ordered sequences. Anyway, it all depends on definitions.
There definitely are not. Rationals are countable, reals are not.
It's tricky to prove things about the sizes of infinite sets, and "there's infinitely many of them interleaved with each other" doesn't do it, at least to my knowledge. I don't know of a way to prove equipotence except by creating a 1-1 relationship between the sets in question.