That's weird. I thought that it was proven that existence of sets larger than aleph-0 but smaller than the number of real numbers is undecidable and you can add it (or negation of it) as additional axiom to math.

https://en.wikipedia.org/wiki/Cardinality_of_the_continuum

"The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between aleph-0 and c=aleph-1. The truth or falsity of this hypothesis is undecidable and cannot be proven within the widely used ZFC system of axioms."