Do mathematicians ever prove that something can or can't be proved? I was just idly thinking about things people have a hard time proving, like P=NP, etc, and wondering if instead it could be proved that it's provable or unprovable.
Is that a thing? Does that ever happen?
 A: It is something mathematicians sometimes do, particularly set theorists and logicians. There is, of course, Goedel's result that no theory can prove its own consistency. There are also results that exploit this to make other provability statements. An example from my own field is that New Foundations set theory can't prove that something called the Axiom of Counting, because if it did it would prove the consistency of NF; we conclude that it is either inconsistent with NF, or a proper strengthening of the theory. Similarly, independence results are exactly theorems stating that neither a statement nor its negation can be proven from a set of axioms.
Strangely, I can't think of any examples of proving that something is provable that don't actually involve proving it, though in principle there's no reason that couldn't happen.
A: A simple example.  In the theory of groups, the proposition
$$
\forall x\;\forall y\quad xy=yx
$$
cannot be proved.  The way to show this is to exhibit a nonabelian group.
In fact, when we say "can be proved" or "cannot be proved" we mean in a certain theory.  And it makes no sense to say it without somehow having that theory in mind.
