Is there any statement unprovable in ZFC without the Continuum Hypothesis and provable with it, which is not (unprovable in ZFC with the "weakened continuum hypothesis" and provable with it):
In which the weakened continuum hypothesis is as follows: "The existence of any set whose cardinality is strictly between that of the integers and the real numbers, is independent of ZFC".
I am probing at the notion of whether the continuum hypothesis is an "ungrounded statement". In particular I suspect that proofs which depend upon the continuum hypothesis are, in reality depending on the weaker property of its "truth or otherwise" being independent of ZFC, rather than depending upon its "truth or otherwise" itself.
Sorry - an edit: when I ask for an unprovable/provable statement, I mean some meaningful statement about the numbers such as the existence of a solution to some diophantine equation.