February 19
James Geiser,
Soundness and the Gödel Undecidability Theorem
The goal of Gödel’s argument that the theory (T) of Peano Arithmetic is not complete, was to show that the Gödel sentences, $G$ , and it’s negation, are not provable in T, unless T is inconsistent. In this paper we examine the first half of this argument, namely, that from a hypothetical derivation, $P_{G}$, of $G$, a derivation, $P_{f}$, can be constructed that ends in a contradiction. We make the observation that the Gödel argument depends on the metatheory concept of representability that, in turn, depends on the metatheory concept of soundness. Our analysis leads to two main observations, the first well know, and the second, a challenge to the standard undecidability argument.
1 – The existence of $P_G$ implies that T is unsound. This conclusion does not require the further construction, from $P_G$, of the derivation $P_f$.
2 - We argue that effectuation of the construction of $P_f$ is obstructed, because that effectuation requires acceptance of a contradiction in the metatheory regarding the soundness of T.
This is joint work with Catherine Hennix.