  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 $\Pi_n$-sentences or equivalently the $\Sigma_n$-soundness of the
theory, and the other is $n$-consistency the restriction of
$\omega$-consistency to the $\Sigma_n$-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
$\Sigma_n$-soundness assumption is constructive, it is shown that there is no
constructive proof for the incompleteness theorem using the $n$-consistency
assumption, for $n\!>\!2$.
