TITLE: Troubles with continuum hypothesis
QUESTION [6 upvotes]: 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?)

REPLY [3 votes]: I posted this previously, but am unable to find it:
There is a theorem by Erdös, that may be relevant to you:
The following are equivalent

$CH$
There is an uncountable family $\mathcal F$ of entire functions, such that for all $z \in \mathbb C$ the set $\{ f(z) \mid f \in \mathcal F \} $ is countable.