**MW:** It’s been ages since John Baez and I discussed Enayat’s paper—not since October 2020! John has since moved on to fresh woods and pastures new. I’ve been reading novels. But I feel I owe it to our millions of readers to finish the tale, so here goes.

Since it has been awhile, I’ll start with a recap. Last season on Nonstandard Models of Arithmetic…

Our goal was to understand Enayat’s paper, Standard Models of Arithmetic. The very title seems a bit paradoxical—nonstandard standard models of arithmetic?!? But by “standard”, Enayat means “the ‘standard’ (nonnegative) integers ω in a model of ZF set theory”.

Since John has been duking it out with Plato’s ghost for decades, and I like to pretend to be a Platonist, it took us a little while just to agree on terminology. We finally settled on the “three decker sandwich” paradigm, whose bumper-sticker reads

ω* ^{U}*⊂

*U*⊂

*V*.

Here *V* is the “real” universe of all sets, where “real” just means “we won’t be looking *outside* of *V*”. *U* is a model of ZF sitting inside *V*. And *ω ^{U}* is what

*U*“thinks is the real ω”. Since

*U*is not the “real” universe of all sets, its ω need not be the “real” ω (the ω of

*V*). In fact,

*ω*must be an initial segment of

^{V}*ω*. That’s because ZF can prove that ω is an initial segment of

^{U}*any*model of PA.

(We assume of course that *V* satisfies ZF. And since *V* is the outermost universe, I’ll usually just write ω instead of *ω ^{V}*.)

Enayat says that *ω ^{U}* is

*ZF-standard*. Most important:

*ω*may be

^{U}*nonstandard*so far as

*V*is concerned. In that case of course,

*U*must look pretty strange to

*V*. If

*U*is a model of some extension

*T*of ZF, Enayat says that

*ω*is

^{U}*T*-standard.

Enayat’s question: can we characterize *T*-standard models of PA, for a recursively axiomatizable *T* extending ZF? Enayat proves a couple of big results that go far towards an answer. These both mention *recursive saturation*, a concept that occupied us for several posts. While recapping this, I’ll dot some i’s and cross some t’s.

We start with the concept of an *n*-type for a model *N* of PA. (We first encountered these in post 1 and post 2.) If *a*_{1}*,…,a _{n}*∈

*N*, then the

*n*-type of (

*a*

_{1}

*,…,a*) is the set of all formulas φ(

_{n}*x*

_{1}

*,…,x*) in L(PA) such that

_{n}*N*⊧φ(

*a*

_{1}

*,…,a*). Or actually, since we want to allow parameters:

_{n}- The
*n*-type of (*a*_{1}*,…,a*) with parameters_{n}*c*_{1}*,…,c*∈_{m}*N*is

{φ(*x*_{1}*,…,x _{n}*,

*y*

_{1}

*,…,y*)∈L(PA):

_{m}*N*⊧φ(

*a*

_{1}

*,…,a*,

_{n}*c*

_{1}

*,…,c*)}.

_{m}- Because of the recursive pairing mechanism of PA, we can restrict ourselves to one free variable and one parameter: {φ(
*x,y*)∈L(PA) :*N*⊧φ(*a*,*c*)}. In other words, any*n*-type can be “coded” as a 1-type. This fact will make our notation less cluttered. I’ll just write “type” from now on.

These are the *realized* types. What about types more generally? First recall that Th_{N}(*N*) is the set of all sentences in L_{N}(PA) that hold in *N*, where L_{N}(PA)_{ }is L(PA) augmented with a name for every element of *N*. (Th_{N}(*N*) is called the *elementary diagram* of *N*.) Note that Th_{N}(*N*) is a complete theory. Also, every model of Th_{N}(*N*) contains an isomorphic copy of *N*.

- A
*type over N*is a set of formulas in L_{N}(PA), {φ(_{i}*x*,*c*) :*i*∈ℕ}, that is consistent with Th_{N}(*N*) in this sense: if*a*is a brand-new constant, then Th_{N}(*N*)+{φ(_{i}*a*,*c*) :*i*∈*I*} is a consistent theory. - (Here
*c*is supposed to be the only name from*N*, so all the φ(_{i}*x*,*y*) belong to L(PA).) - (If the type has only finitely many different formulas, just repeat one of them endlessly.)
- Equivalently, by the Completeness Theorem,
*N*has an extension*M*containing an element*a*such that*M*⊧φ(_{i}*a*,*c*) for all φ. The element_{i}*a*realizes the type in*M*.

It’s useful to think of a type as an infinite conjuction:

- Think of {φ
(_{i}*x,c*) :*i*∈ℕ} as ⋀_{i∈ℕ}φ(_{i}*x,c*). - Because Th
_{N}(*N*)+{φ(_{i}*a*,*c*) :*i*∈ℕ} is consistent iff every finite subset is, we have another version of the consistency requirement: Th_{N}(*N*)+{φ(_{i}*a*,*c*) : 1≤*i*≤*k*} is consistent for every*k*. - Because
*a*is a brand-new constant, yet another: Th_{N}(*N*)+∃*x*⋀_{1≤i≤k}φ(_{i}*x*,*c*) is consistent for every*k*. - Because Th
_{N}(*N*) is complete, yet one more version: Th_{N}(*N*)⊢∃*x*⋀_{1≤i≤k}φ(_{i}*x*,*c*) for every*k*. - Or finally, this version:
*N*⊧∃*x*⋀_{1≤i≤k}φ(_{i}*x*,*c*) for every*k.*

After all that throat-clearing, we’re ready for the notion of *saturated*:

*N*is*saturated*if every type that is realized in some elementary extension of*N*(i.e., model of Th_{N}(*N*)) is already realized in*N*.- Equivalently: given a set of formulas {φ
(_{i}*x,c*) :*i*∈ℕ} as above, if for every*k*we have*N*⊧∃*x*⋀_{1≤i≤k}φ(_{i}*x,c*), then*N*contains an element*a*such that*N*⊧φ(_{i}*a,c*) for all*i*∈ℕ.

For *recursively saturated* we need the concept of a *recursive type*:

- A type {φ
(_{i}*x,c*) :*i*∈ℕ} over*N*is*recursive*when the set of Gödel numbers {⌜φ(_{i}*x,y*)⌝ :*i*∈ℕ} is a recursive set. - Notice that all the φ
(_{i}*x,y*) belong to L(PA). We sidestep the question, what would “recursive” even mean for non-standard parameters? We begin with a recursively given collection of formulas in L(PA), and then plug the parameter in. *N*is*recursively saturated*if every recursive type is realized in*N*.

That’s quite a lot to chew on. I’ll finish the recap in the next post.