Can we prove that a statement cannot be proved? Is it possible to prove that you can't prove some theorem? For example: Prove that you can't prove the Riemann hypothesis. I have a feeling it's not possible but I want to know for sure. Thanks in advance

As far as I know the continuum hypothesis has been proved "independent" from the ZFC axioms. So you can assume it true or false.
This is very different from just true undecidible statements (as the one used by Gödel to prove his incompleteness theorem). Gödel statements is true even though it is not derived by the axioms described in the theorem (Peano axioms) but it is proved by a different argument (the theorem itself).
