Roughly speaking, the Gödel statement, G, asserts: "G cannot be proven true". If G were able to be proven true under the theory's axioms, then the theory would have a theorem, G, which contradicts itself, and thus the theory would be inconsistent. But if G were not provable, then it would be true (for G expresses this very fact) and thus the theory would be incomplete.
(i.e.. G = there is no even number greater than 2 that is not divisible by 2)
The argument just given is in ordinary English and thus not mathematically rigorous. In order to provide a rigorous proof, Gödel represented statements by numbers; then the theory, which is already about numbers, also pertains to statements, including its own. Questions about the provability of statements are represented as questions about the properties of numbers, which would be decidable by the theory if it were complete. In these terms, the Gödel sentence is a claim that there does not exist a natural number with a certain property. A number with that property would be a proof of inconsistency of the theory. If there were such a number then the theory would be inconsistent, contrary to hypothesis. So, assuming the theory is consistent (as done in the theorem's hypothesis) there is no such number, and the Gödel statement is true, but the theory cannot prove it. An important conceptual point is that we must assume that the theory is consistent in order to state that this statement is true.
Consider this mathematical formula T(A(a))=O. Let
m = Code[ T(A(a))=O ].
Consider the statement below.
T(A(m))=O
Contemplate the fact that the statement above is a real mathematical statement in set theory because the functions T and A are constructible in set theory and the natural number m is also precisely defined.
Notice that
(*) A(m) = Code[ T(A(m))=O ].
Suppose that T(A(m))=O is a theorem. Then by the property of function T, A(m) is not an encoded theorem. However, by (*) A(m) encodes T(A(m))=O, which is a theorem. Contradiction.
So we showed that T(A(m))=O is not a theorem.
Suppose that ~(T(A(m))=O) is a theorem. Hence T(A(m))=1 is a theorem. So by the property of function T, A(m) is an encoded theorem. However, by (*) A(m) encodes T(A(m))=O. So T(A(m))=O is a theorem. But we already had that ~(T(A(m))=O) is a theorem. Since we believe that axiomatic set theory is consistent, we have a contradiction.
So we showed that ~(T(A(m))=O) is not a theorem.
We have shown that
T(A(m))=O is not a theorem.
~(T(A(m))=O) is not a theorem.
So we have finally constructed a statement in set theory such that it is not a theorem and its negation is not a theorem.
In order to solve the dilemma presented by the Goedel theory, I believe that it would need an anti-theory that produces statements that directly contradict the truths derived from the Goedel axioms. A direct contradiction would mean a truth. An agreement would mean a false. Which would be understood as a number unaccountable in the original Goedel theory, but since it is now accounted for, would make the theory complete, and since both the theory and the anti-theory are part of the overarching theory, it would be consistent. Thus, a computer programmed to answer mathematical questions from its programmed set of axioms: G is True. G = the computer will say that G is false. Ask computer, is G true. The computer will say false. The anti-Goedel theory will say that G is False. G = the computer will say that G is false. Is G true or false, the computer will say false.
So you have computer and lie-computer
G = X
computer, does G = X
true
G = X
lie-computer, does G = X
false