Eliezer wrote: “Godel’s Completeness theorem shows that any first-order statement true in all models of a set of first-order axioms is provable from those axioms. Thus, the failure of Peano Arithmetic to prove itself consistent is because there are many “supernatural” models of PA in which PA itself is not consistent; that is, there exist supernatural numbers corresponding to proofs of P&~P.”
This is getting far from the topic but… I really don’t see how Completeness entails anything about PA’s failure to prove itself consistent (much less how it suggests an explanation in terms of “supernatural models”, whatever that is supposed to mean). PA is not expressible as a first-order statement, so Completeness has nothing to say about PA or its limitations.
Eliezer wrote: “Godel’s Completeness theorem shows that any first-order statement true in all models of a set of first-order axioms is provable from those axioms. Thus, the failure of Peano Arithmetic to prove itself consistent is because there are many “supernatural” models of PA in which PA itself is not consistent; that is, there exist supernatural numbers corresponding to proofs of P&~P.”
This is getting far from the topic but… I really don’t see how Completeness entails anything about PA’s failure to prove itself consistent (much less how it suggests an explanation in terms of “supernatural models”, whatever that is supposed to mean). PA is not expressible as a first-order statement, so Completeness has nothing to say about PA or its limitations.