Do we know if there exist true mathematical statements that can not be proven? 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?

Amongst the many excellent answers you have received, nobody appears to have directly answered your question.
Goldbach's conjecture can be true and provable, true but not provable using the "normal rules of arithmetic", or false. 
There are strong statistical arguments which suggest it is almost certainly true. 
Whether it is provable using the "normal laws of arithmetic" - like those used to prove Fermat's Last Theorem or the Prime Number Theorem and everything you learned in high school maths - is not known. Assuming it can't be proven is a complete dead-end. To be interested at all you have to either assume it is true and be looking for a proof, or assume it is false and be looking for a counter example.
