It’s sometimes thought or said that the second incompleteness theorem shows that formal systems such as PA are in some sense “incapable of self-reflection”, and their incapability to prove their own consistency reflects this lacuna in their “cognitive capabilities” or something along those lines.

Löb’s theorem highlights how this is a much deeper phenomenon: it’s not just that these formal systems lack abilities, or formalizations of relevant mathematical insight, or whatever; no, it’s that it’s formallyimpossible for any consistent system of the relevant sort to even baldly assert their own consistency, on pain of inconsistency!

That is, we can for instance construct, using the diagonal lemma, a sentence A such that A is, provably in PA, equivalent to “PA + A is consistent”—thus creating, in a sense a theory that just asserts its own consistency, in the form of a random assertion about Diophantine equations or however we carry out the arithmetization of syntax, for no particular legible reason whatever—but we then find that PA + A is inconsistent! (This is an immediate consequence of the second incompleteness theorem.)

I recommend Torkel Franzén’s excellent “Inexhaustibility—a non-exhaustive treatment” for anyone pondering these matters.

It’s sometimes thought or said that the second incompleteness theorem shows that formal systems such as PA are in some sense “incapable of self-reflection”, and their incapability to prove their own consistency reflects this lacuna in their “cognitive capabilities” or something along those lines.

Löb’s theorem highlights how this is a much deeper phenomenon: it’s not just that these formal systems lack abilities, or formalizations of relevant mathematical insight, or whatever; no, it’s that it’s

formallyimpossiblefor any consistent system of the relevant sort to evenbaldly asserttheir own consistency, on pain of inconsistency!That is, we can for instance construct, using the diagonal lemma, a sentence A such that A is, provably in PA, equivalent to “PA + A is consistent”—thus creating, in a sense a theory that just asserts its own consistency, in the form of a random assertion about Diophantine equations or however we carry out the arithmetization of syntax, for no particular legible reason whatever—but we then find that PA + A is inconsistent! (This is an immediate consequence of the second incompleteness theorem.)

I recommend Torkel Franzén’s excellent “Inexhaustibility—a non-exhaustive treatment” for anyone pondering these matters.