Hrm… if the system is isn’t necessarily trustworthy, then that the system proves that it can prove P doesn’t mean that it’s actually true that it can prove P, I guess.
EDIT: actually, having it as an explicit axiom “If this proves P, then P” runs you into trouble in any system that has something like Lob’s theorem.
(“if some specific subset of the rest of this system, (ie, other than this axiom) proves P, then P” can potentially be okay, though)
Hrm… if the system is isn’t necessarily trustworthy, then that the system proves that it can prove P doesn’t mean that it’s actually true that it can prove P, I guess.
EDIT: actually, having it as an explicit axiom “If this proves P, then P” runs you into trouble in any system that has something like Lob’s theorem.
(“if some specific subset of the rest of this system, (ie, other than this axiom) proves P, then P” can potentially be okay, though)