TITLE: Do we know if there exist true mathematical statements that can not be proven?
QUESTION [164 upvotes]: Given the set of standard axioms (I'm not asking for proof of those), do we know for sure that a proof exists for all unproven theorems? For example, I believe the Goldbach Conjecture is not proven even though we "consider" it true.
Phrased another way, have we proven that if a mathematical statement is true, a proof of it exists? That, therefore, anything that is true can be proven, and anything that cannot be proven is not true? Or, is there a counterexample to that statement?
If it hasn't been proven either way, do we have a strong idea one way or the other? Is it generally thought that some theorems can have no proof, or not?

REPLY [2 votes]: Let $\Sigma$ be Rado's Busy Beaver function. 
There exist specific positive integers $k$ and $n$ such that $"\Sigma(k) = n"$ is true-but-unprovable in ZFC. 
Also see this question and answer.