okay is there anyone that knows more provability stuff than me that could take this question?
For FairBot, we have
PA ⊢ F(X) ↔ □X(F)
... right?
This seems obvious. It's a specialization of the model agent definition on page 11 of the paper to φ(q)=□q.
But like... okay, we've got the FairBot definition as a program on page 7... then we construct some PA formula that behaves like that. And so we must have a theorem of PA saying this is the behavior, right?
How to, like, say this.
Or is the disconnect that he missed that we then also have
PA ⊢ □(F(X) ↔ □X(F))
?
(medium confidence:) I think this is probably just the fact that the "modal" setup already abstracts away a bit from actual computer programs. Like, the program on page 7
can't actually be written, because the question of whether PA proves some formula or not is undecidable. But if you have such a subroutine, then you could easily prove that the program works according to the forumula, and then you could further encode that proof as a an object in PA (the necessitation rule). So I think the issue is not really how to add the extra box, but more how to write the program/prove the boxless formula in the first place.

















