I have two question regarding Godel's incompletness theorem. The theorem says that every axiomatic system is either incomplete or inconsistent. If it's consistent, then there are true statements that are unprovable.
If so, then it must also be the case that there are false statements that cannot be proven to be false, becuase if we could prove that every false statement is false, then we would be able to prove every true statement - by contradiction.
If every false statement was provably false, then I could just take the unprovable true statement $A$, prove the falsehood of its negation and I would have a proof of $A$.
Then there must exist both true and false statements that cannot be proven true and false respectively.
Question: where is the logical error I'm making here?