Let $S$ be some statement which is unprovable but true in an axiomatic system $T$. If $T$ is consistent, then adding $S$ as an axiom of $T$ keeps the system consistent. But what about adding $\neg S$ as an axiom?
For example, the continuum hypothesis is unprovable in ZFC, and we can add it or its negation as an axiom with no problem.
However, if the Goldbach conjecture is unprovable, it must be true, since if it were false we'd be able to find a counter-example, and check that it is a counter-example. Hence we wouldn't be able to add the negation of the Goldbach conjecture as an axiom if it is unprovable. Does this imply it is provable? Or is this okay?