Does Gödel's incompleteness theorem invoke a Law of Excluded Middle contradiction?

Question

Does Gödel's incompleteness theorem cause the Law of Non Contradiction to contradict its self?

If so, would this be a considered a conjecture?

Answer 1

No, not really. Gödel's theorem does not appeal to notion of "truth" of a conjecture per se - merely to its provability within a given set of axioms. The "provability" operator can be intuitively understood as $\square_A \psi= A \Rightarrow \psi$, $A$ being the set of axioms: even under the law of excluded middle, it is not guaranteed that $[\lnot(\xi \Rightarrow \psi)] \Rightarrow (\xi \Rightarrow \lnot \psi)$.

Answer 2

The fact that, by Gödel's incompleteness theorem,

"there will always be statements about the natural numbers that are true, but that are unprovable within the system"

does not contradict the Law of Excluded Middle. It means that there is a statement $G$ about natural numbers that is true in the standard model of arithmetic (the one whose domain is the set of natural number $0$, $1$, $2$, $\dots$), but is false in non-standard models of first-order Peano arithmetic (this is the reason why $G$ is unprovable in first-order Peano arithmetic). Indeed, according to Gödel's completeness theorem for first-order logic, a statement is provable in a system if and only if it is true in every model of such a system: this not the case for the statement $G$ and Peano arithmetic as system.