TITLE: Statement that is provable in $ZFC+CH$ yet unprovable in $ZFC+\lnot CH$
QUESTION [11 upvotes]: My understanding of logic is really basic, and I ask this question out of curiosity.
Is there an explicit example of a statement whose proof uses the continuum hypothesis and is unprovable in $ZFC + \lnot CH$. 
ADDED: Preferably I'm looking for an example outside set theory.

REPLY [3 votes]: There exists a subset $S$ of plane $\mathbb{R}^2$, such that intersection of any horizontal line and $S$ is countable, and intersection of any vertical line with $S$ has countable complement.
It's quite easy to show that if such $S$ exists, then $\aleph_1 = \mathfrak{c}$, so it's not provable in ZFC + $\lnot$ CH. It's a bit harder to show that if CH holds, then such $S$ exists.