Some stuff about models of PA+~Con(PA)

Assume Peano Arithmetic (PA) is consistent. Then it can’t prove its own consistency. Thus, there is a model M of PA according to which PA is inconsistent, and hence, according M, there is a proof of a contradiction from a finite set of axioms of PA. This sounds very weird.

But it becomes less weird when we realize what these claims do and do not mean in M.

The model M will indeed contain an M-natural number a that according to M encodes a finite sequence of axioms of PA, and it will also contain an M-natural number p that according to M encodes a proof of a contradiction using the axioms encoded in A.

However, here are some crucial qualifications. Distinguish between the M-natural numbers that are standard, i.e., correspond to an actually natural number, one that from the point of view of the “actual” natural numbers is finite, and those that are not. The latter are infinite from the point of view of the actual natural numbers.

First, the M-natural number a is non-standard. For a standard natural number will only encode a finite number of axioms, and for any finite subtheory of PA, PA can prove its consistency (this is the “reflexivity of PA”, proved by Mostowski in the middle of the last century). Thus, if a were a standard natural number, according to M there would be no contradiction from the axioms in a.

Second, while every item encoded in a is according to M an axiom of PA, this is not actually true. This is because any M-finite sequence of M-natural numbers will either be a standardly finite length sequence of standard natural numbers, or will contain a non-standard number. For let n be the largest element in the sequence. If this is standard, then we have a standardly finite length sequence of standard natural numbers. If not, then the sequence contains a non-standard number. Thus, a contains something that is not axiom of PA.

In other words, according to our model M, there is a contradictory collection of axioms of PA, but when we query M as to what that collection is, we find out that some of the things that M included in the collection are not actually axioms of PA. (In fact, they won’t even be well-formed formulas, since they will be infinitely long.) So a crucial part of the reason why M disagrees with the “true” model of the naturals about the consistency of PA is because M disagrees with it about what PA actually says!