This is a misunderstanding. Statement A is actually true in the system, you just cannot prove that it is true. Adding an axiom specifying its falsity would be a contradiction (although you could not prove this).
Adding either the Godel sentence or its complement would ruin the consistency of the axiomatic system, because the whole point of the Godel sentence is that it claims that itself cannot be proven to be true in its axiomatic system. But you don't get to add axioms to an axiomatic system anyway, because doing so yields a new axiomatic system to which the original Godel sentence does not refer.
I think by adding the axiom, they mean considering the axiom system with the old axioms, and also the one being added?
So, assuming the initial system is consistent, adding the godel sentence, or its negation, produces a new consistent system, iirc. But in the second case, it would be omega-inconsistent ?
