Perhaps we have chosen the wrong definition? Anyway I'm basically just wondering if this is still an open question in mathematics or if it has already been proven or disproven. Are the "sets" of these cardinalities something specific that we know to exist, or just something that exists merely...
For the infinite cardinal numbers (of which there are infinitely many), do they each necessarily correspond to some set? I mean we know that aleph-naught corresponds to N, c (aleph-one by continuum hypothesis) corresponds to R, but what about all the other infinite cardinals?
Is it possible...