He showed that a structure of axioms will not be able to prove or disprove all possible > theorems.

Not quite. Godel's theorems are a bit more subtle. The first incompleteness theorem says that any consistent, complete axiomatic system which can model the natural numbers must contain statetements which cannot be proved or disproved in that system. (Even that is a bit of an imprecise statement. One needs to be careful about what one means by an axiomatic system. For our purposes, this can be phrased as a set of axioms and rules of inference which can be listed and applied using an effective method).

The second, closely related theorem, is that any axiomatic system obeying the conditions of the first theorem contains a statement equivalent to its own consistency iff the system is inconsistent. This also needs a fair bit of unpacking. The rough idea here is that any axiomatic system that can contain the natural numbers and is powerful enough to talk about an analog of itself cannot prove that analog's consistency.