Suppose we had a model M that we thought described cannons and cannon balls. M consists of a set of mathematical assertions about cannons

In logic, the technical terms 'theory' and 'model' have rather precise meanings. If M is a collection of mathematical assertions then it's a theory rather than a model.

formally independent of the mathematical system A in the sense that the addition of some axiom A0 implies Q, while the addition of its negation, ~A0, implies ~Q.

Here you need to specify that adding A0 or ~A0 doesn't make the theory inconsistent, which is equivalent to just saying: "Neither Q nor ~Q can be deduced from A."

Note: if by M you had actually meant a model, in the sense of model theory, then for every well-formed sentence s, either M satisfies s or M satisfies ~s. But then models are abstract mathematical objects (like 'the integers'), and there's usually no way to know which sentences a model satisfies.