$\begingroup$

ZFC is consistient with the negation of the continuum hypothesis. Do theories that deny the continuum hypothesis have any use outside of metamathematics at the moment?

For example, I know that theories that deny the axiom of choice enjoy use in constructive mathematics and category theory.

I know that one cool thing you can do with cardinality is easily prove that something without a given property exists by showing that the total number of things is more than the number of things with that property (irrational number, transcendent numbers, uncomputable numbers, although I think these examples where first proven via other methods). Adding more cardinal numbers would probably help with this.