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

Godel’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 ${\mathrm{\Pi }}_n$-sentences or equivalently the ${\mathrm{\Sigma }}_n$-soundness of the theory, and the other is $n$-consistency the restriction of $\omega$-consistency to the ${\mathrm{\Sigma }}_n$-formulas. It is also shown that Rosser’s Incompleteness Theorem does not generally hold for definable non-recursively enumerable theories, whence Godel-Rosser’s Incompleteness Theorem is optimal in a sense. Though the proof of the incompleteness theorem using the ${\mathrm{\Sigma }}_n$-soundness assumption is constructive, it is shown that there is no constructive proof for the incompleteness theorem using the $n$-consistency assumption, for $n\text{\hspace{0.17em}⁣}>\text{\hspace{0.17em}⁣}2$.