MachuPikacchu on Nostr: There are! Once you allow the axiom of infinity you end up with an infinite number of ...

There are! Once you allow the axiom of infinity you end up with an infinite number of distinct infinities actually; we say that they each have different "cardinality." I had to prove all sorts of statements about infinite sets in a topology course I took years ago but I'm rusty at this point.

A set is said to be infinite if there exists a bijection between itself and a proper subset. So for example, there's a 1-1 mapping between the set of integers and the set of even integers: for each integer multiply it by 2 and you get the evens; for each even integer divide it by 2 and you get the whole set of integers again. That doesn't work for any finite set.

And suppose you have an infinite set, S. Then the set of all subsets of S (called the power set of S) actually has a higher cardinality than S itself and there doesn't exist a bijection between S and its power set even though both are infinite.

However, again all that supposes you accept the concept of an infinity in the first place because only then can you construct all those things. Modern math assumes it as a given.