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Godel's First Incompleteness Theorem is generalized to definable theories,which are not necessarily recursively enumerable, by using a couple ofsyntactic-semantic notions, one is the consistency of a theory with the set ofall true $\Pi_n$-sentences or equivalently the $\Sigma_n$-soundness of thetheory, and the other is $n$-consistency the restriction of$\omega$-consistency to the $\Sigma_n$-formulas. It is also shown that Rosser'sIncompleteness Theorem does not generally hold for definable non-recursivelyenumerable theories, whence Godel-Rosser's Incompleteness Theorem is optimal ina sense. Though the proof of the incompleteness theorem using the$\Sigma_n$-soundness assumption is constructive, it is shown that there is noconstructive proof for the incompleteness theorem using the $n$-consistencyassumption, for $n\!>\!2$.

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