Comment by Kotlopou

Asker of the SO question here: I edited the question to link to a few other answers on this kind of thing. Essentially, "PA is consistent" is not enough, but a "uniform reflection principle" that says "if PA proves something, it's true" is enough. I'm not 100% certain that this principle is equivalent to omega-consistency, but if I'm reading this correctly, it should be:

https://en.wikipedia.org/wiki/%CE%A9-consistent_theory#Relat...

The Wikipedia article says T is omega-consistent if "T + RFN_T + the set of all true sentences is consistent", which should mean the same thing as "T + RFN_T is true".