Godel's second incompleteness theorem for Sigma(n)-definable theories

Godel's second incompleteness theorem is generalized by showing that any Sigma(n+1)-definable and Sigma(n)-sound extension of Peano arithmetic (PA) cannot prove its own Sigma(n)-soundness. The optimality of the generalization is shown by presenting a Sigma(n+1)-definable and Sigma(n-1)-sound extension of PA that proves its own Sigma(n-1)-soundness.