Godel-Rosser's Incompleteness Theorems for Non-Recursively Enumerable Theories

Abstract

Gödel's First Incompleteness Theorem is generalized to definable theories, which are not necessarily recursively enumerable, by using a couple of syntactic-semantic notions; one is the consistency of a theory with the set of all true Π n -sentences or equivalently the Σ n -soundness of the theory, and the other is n-consistency the restriction of ω-consistency to the Σ n -formulas. It is also shown that Rosser's Incompleteness Theorem does not generally hold for definable non-recursively enumerable theories; whence Gödel-Rosser's Incompleteness Theorem is optimal in a sense. Though the proof of the incompleteness theorem using the Σ n -soundness assumption is constructive, it is shown that there is no constructive proof for the incompleteness theorem using the n-consistency assumption, for n > 2.