Axiom of choice is discussed very often, because it should be a lot of paradoxes (Banach-Tarski paradox, for example) and in general it is considered by many non-obvious (for uncountable case, of course). But continuum hypothesis remains aside; it doesn't seem so strange. How many people — so many opinions. But so many theorems only from calculus use AC; it seems that it is easier to accept than reject AC.
And what about CH? There was a question on SE, but I would like to know what the (interesting) theorems using the CH; I don't know any such theorem. Math with AC and math without AC seems to me very different; but math with and without CH... Who cares?)