after reading this https://en.wikipedia.org/wiki/Cardinality_of_the_continuum
I wonder what good is it as a tool in mathematics if there is no proof. is it merely useful to prove that it is undecidable? wikipedia says no proof in ZFC but I wonder if there are other axiom systems that may be more or less strict where this problem may actually be decidable. for example, is there some weaker result that can be gained from homotopy type theory where there is some type with no distinction between different sets with the same cardinality? sorry if these are dumb questions. I still don't understand a lot of this but I am desperately trying to read as much as I can. I did not find this question asked exactly like this through search on this forum.