Godel's Second Incompleteness Theorem for Definable Theories

Abstract

It is proved that if $T$ is a $\Sigma_{n+1}$ Definable theory which is $\Sigma_n$-sound and extends $PA$, then $T$ can not prove the sentence $\Sigma_n-sound(T)$ that expresses the $\Sigma_n$-soundness of $T$. Optimality of this result is showed by constructing a $\Sigma_{n+1}$-definable and $\Sigma_{n-1}$-sound theory extending $PA$ such that $\Sigma_n-sound(T)$ is $T$-provable. It is also proved that no R.E. arithmetical theory, evevn very weak theories which are not $\Sigma_1$-complete, can prove $\Sigma_1$-soundness of itself.