Non-standard Models of Arithmetic 1

This is the first in a series of posts, recording an e-conversation between John Baez (JB) and me (MW).

JB: I’ve lately been trying to learn about nonstandard models of Peano arithmetic. Do you know what a “recursively saturated” model is? They’re supposed to be important but I don’t get the idea yet.

MW: What books and/or papers are you reading? I used to know this stuff, indeed my thesis (1980) was on existentially complete models of arithmetic. When I looked at it a couple of years ago, I was amazed at how much I’d forgotten. Talk about depressing.

Anyway, I’ll toss out a few vague ideas, to see if they help. Maybe this will be the push I need to get back to Kaye’s book, or even Kossak & Schmerl. I picked them up a few months ago, hoping to revisit my youth, but I didn’t make it past the prefaces.

As Hodges puts it, model theory is “algebraic geometry minus fields”. If you have an algebraic number r in a extension field K/F, it’s natural to look at all the polynomials in F[x] which have r as a root. It turns out that this is a principal ideal, generated by the minimal polynomial.

Sort-of generalize to an arbitrary model M of a theory T. Let $(r_1,\ldots,r_n)$ be an n-tuple of elements of M. Look at the set of all formulas $\varphi(x_1,\ldots,x_n)$ such that M satisfies $\varphi(x_1,\ldots,x_n)$. This is the complete n-type of $(r_1,\ldots,r_n)$ with respect to M.

Unlike the case in field theory, complete n-types are not usually implied (“generated”) by a single formula, but when they are, they are called principal.

Next step is to free the notion of n-type from dependence on the model M. If we have a set of formulas $\Phi=\{\varphi(x_1,\ldots,x_n)\}$ that is consistent with T, then it’s an n-type. (Sort of like a polynomial in F[x], looking for a root.) The type is realized in a model M if there is a n-tuple $(r_1,\ldots,r_n)$ such that M satisfies all the $\varphi(x_1,\ldots,x_n)$. It’s omitted if there is no such n-tuple in M.

The omitting types theorem says that any countable collection of non-principal n-types in a countable language can all be omitted by some model of T.

You can imagine that n-types tell us a lot about possible isomorphisms and automorphisms of models. The poster child: the theory DLO of dense linear orderings without endpoints. This is because there are essentially only very simple types. Pretty much the only thing you can say about $(r_1,\ldots,r_n)$ is what permutation of the subscripts puts them in monotonically increasing order, and for which subscripts i and j we have ri=rj. Anything else about the n-tuple is implied by this. Especially significant is that no quantifiers need apply.

And of course, any two countable DLOs are isomorphic, and there are lots of automorphisms of a countable DLO. If you look at the back-and-forth argument, you’ll see it relies on the limited repetoire of types.

Also, to hammer the analogy with field theory, the polynomials r satisfies tells us all about the isomorphisms and automorphisms.

OK, now specialize to PA. Key thing here is the overspill lemma. This allows you to code info about subsets of the standard $\mathbb{N}$ into single elements of a non-standard M. For example, the set of standard prime divisors of a non-standard r in M.

If the set you want to code is totally arbitrary, well, you can’t even express it in the language of PA, so we want to look at sets that are definable by a formula. Then we can apply the overspill lemma. Say φ(x) is the formula for the set. Consider the formula $\psi(y)\equiv(\exists z)(\forall x. Here px is the x-th prime. Since ψ(n) holds for arbitrarily large finite n‘s (indeed all finite n‘s), overspill says that it also holds for some non-standard n. So there is a z such that φ(x) is true iff px|z, for all x<n. In particular it holds for all finite x, and so z codes the set via its prime divisors.

More generally, it would be nice to look at sets of n-tuples defined by an n-type. But we need some way to describe the n-type in the language of PA—we can’t describe an arbitrary set of formulas in the language of PA. So we look at recursive n-types, i.e., sets of formulas whose Gödel’s numbers form a recursive set.

If we’re lucky, we’ll be able to import a lot of classical model theory, developed in ZF (or even naive set theory) into PA, because we can code it all in PA. And this approach will help us understand things like extensions of one model by another, automorphisms, etc.

A couple more notes. First, the prenex normal form hierarchy shows up all the time in logic. A variant especially adapted to PA (the arithmetic hierarchy) is to let $\Delta_0$ formulas allow arbitrary bounded quantifiers, like $(\forall x. It turns out that a relation is recursively enumerable iff it is $\Sigma_1$. So that’s a nice bridge between the model theory of PA and computability theory.

It also turns out that, because of the MRDP theorem, you don’t even need to allow bounded quantifiers at the ground level. Well, that’s a bit sloppy. What I mean is that $\Sigma_1$ is the same as $\exists_1$, formulas you get from putting a string of existential quantifiers in front of a quantifier-free formula.