It's also important to point out that if we cant prove a statement or its negation (one of which must be true) [...] Is that true, could it not be neither, i.e

It's also important to point out that if we cant prove a statement or its negation (one of which must be true) [...] Is that true, could it not be neither, i.e. independent of the axioms? Or is this assuming completeness which rules out independent statements?