Godel's Second Incompleteness Theorem for Definable Theories

It is proved that if $T$ is a ${\mathrm{\Sigma }}_{n+1}$ Definable theory which is ${\mathrm{\Sigma }}_n$-sound and extends $PA$, then $T$ can not prove the sentence ${\mathrm{\Sigma }}_n-sound(T)$ that expresses the ${\mathrm{\Sigma }}_n$-soundness of $T$. Optimality of this result is showed by constructing a ${\mathrm{\Sigma }}_{n+1}$-definable and ${\mathrm{\Sigma }}_{n-1}$-sound theory extending $PA$ such that ${\mathrm{\Sigma }}_n-sound(T)$ is $T$-provable. It is also proved that no R.E. arithmetical theory, evevn very weak theories which are not ${\mathrm{\Sigma }}_1$-complete, can prove ${\mathrm{\Sigma }}_1$-soundness of itself.