As the GP points out, that's not what Godel's incompleteness theorem actually shows. Although it's a common misconception (one which unfortunately is propagated by many sources that should know better).

The key point of the incompleteness theorem is that it shows that (at least in first order logic, which is the logic in which the theorem holds) no set of axioms can ever pin down a single model. For example, no set of first-order axioms can ever pin down "the standard natural numbers" as the only model satisfying the axioms. There will always be other models that also satisfy them. So if you want to pin down a single model, you always have to go beyond just a set of first-order axioms.

Using the natural numbers as an example, consider a model that consists of two "chains" of numbers:

(0, 1, 2, 3, ....)

(..., -3a, -2a, -1a, 0a, 1a, 2a, 3a, ...)

The first chain is, of course, the standard natural numbers, but the second chain also satisfies the standard first-order axioms that we normally take to define natural numbers. So this model, as a whole, satisfies those axioms. And there is no way, within first-order logic, to say "I only want my model to include the first chain". That's what Godel's incompleteness theorem (or more precisely, his first incompleteness theorem combined with his completeness theorem) tells us.