  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.
