Let $T$ be an effectively axiomatizable system that we believe to be consistent, and expressive enough so that Godel’s theorem applies to it.
Then we have that the provability statements related to $T$, i.e. the statements of the form $T \vdash \phi$ for some formula $\phi$, are not all decidable inside $T$.
But (unless I missed something) Godel’s theorem does not exclude the possibility of a stronger (but still consistent) effectively axiomatizable system $T’$ that decides all the provability statements related to $T$ : for any $\phi$, $T’$ either proves $T\vdash \phi$ or $T'$ proves $\lnot\big(T \vdash \phi \big)$.
Is any example known of such a pair $(T,T’)$ ?