What does the continuum hypothesis imply?

Question

Are there any fundamental/interesting results that are a consequence of assuming the continuum hypothesis as an additional axiom?

I'm sorry if this question was already asked. I'm also sorry if there is no rigour at all in the way I asked it.

Thanks!

Answer 1

There are many cardinals which have aleph values undetermined in ZFC, but for which we can say that they are either infinite and less than continuum, or uncountable and no greater than continuum, both of which would immediately resolve their (aleph) value if we had CH.

Many of those can be resolved by the somewhat weaker Martin's axiom. It is strictly weaker than CH (if ZFC is consistent, so is ZFC with the axiom and the statement that $\mathfrak c=\omega_2$, but CH implies it by Rasiowa-Sikorski lemma).

For the consequences of Martin's axiom, a good source is David Fremlin's book aptly named "Consequences of Martin's Axiom". Quite many of the results in this book are somewhat trivial if you assume CH, though, if I recall correctly...