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Gödel–Rosser's Incompleteness Theorem is generalized by showingΠ   n  +  1-incompleteness of anyΣ   n  +  1-definable extension of Peano Arithmetic which is eitherΣ  n-sound or    n   -consistent. The optimality of this result is proved by presenting a complete,Σ   n  +  1-definable,Σ   n  −  1-sound, and(   n  −  1   )-consistent theory for anyn  &gt;  0    . Though the proof of the incompleteness theorem forΣ   n  +  1-definable theories using theΣ  n-soundness assumption is constructive, it is shown that there is no constructive proof for the Incompleteness Theorem forΣ   n  +  1-definable theories using the    n   -consistency assumption, whenn  &gt;  2    .

Gödel–Rosser's Incompleteness Theorem, generalized and optimized for definable theories