By Gödel's second incompleteness theorem, no sufficiently powerful formal system can prove its own consistency.
I was wondering what happens if one tries to manually append an axiom stating a formal system's own consistency to an existing formal system.
By the $2$nd incompleteness theorem (assuming the consistency of PA) we can append $\text{Con}(PA)$ to PA to get a consistent formal system which proves (trivially) the consistency of PA. Though this resulting system of course says nothing about itself.
What if instead, we append to $PA$ the following axiom T:
$T:\text{Con}(PA + T)$
The resulting system would seemingly trivially prove its own consistency, having it as an axiom. But also the effort to formally construct this system appears doomed to fail by the $2$nd incompleteness theorem, as it proves its own consistency (and I assume it remains a "sufficiently powerful" formal system as it still contains $PA$).
Where does this go wrong? I assume it is in the attempt to include a self-referencing axiom, though I'm not sure where the issue would lie since formal self-referencing statements aren't intrinsically wrong (as seen in the proof of the $1$st incompleteness theorem).
