"Something I learnt from Mietek Bak is that Löb..."


by Patrick Stevens Jul 6 2016

Something I learnt from Mietek Bak is that Löb's Theorem is kind of more subtle than this. In provability theory, it's fine to have a "box" operator that we informally read as "is provable"; but what Löb's theorem tells us that we can't simply interpret it literally as "is provable" without difficulties. One should define the "provability" predicate formally, to avoid getting confused (or one should specify that it is simply a formal symbol to which we have not assigned any semantic meaning, although that is somewhat against the point of the angle taken by the parent article); for example, the provability predicate could be defined by a certain first-order formula which unpacks a Gödel number, checks it's encoding a proof and verifies each step of the encoded proof.


Jaime Sevilla Molina

Yeah, that is the formal definition of the standard provability predicate. I'll add the page explaining that soon enough.