provability of a mathematical statement

Question

Is it possible to prove that a non-axiom statement is not mathematically provable with current accepted axioms of mathematics?

Note that this is not a question of proving if it is a true or false statement, but a proof that it is impossible to make a statement of the validity of the statement.

for extra clarification: Gödel's incompleteness theorems state that there are true statements that are unprovable. My question is are there statements that cannot have their validity proven.

Answer 1

There are many sentences $\varphi$ that have been shown to be neither provable nor refutable from the axioms of ZFC (if that theory is consistent). The most famous example is the continuum hypothesis. Please see here for a very long list.