limits of stable theories

Question

Say that a complete theory $T$ is a limit of stable theories if for every $\phi \in T$ there is a stable completion of $\{\phi\}$. (Equivalently, $T$ is the ultraproduct of stable theories.)

Question: Is every simple theory a limit of stable theories? (We allow that theories of finite structures are stable.)

---

As motivation, all the simple unstable theories I am aware of are random in various ways (e.g. random graph) and the independence property only comes from the full random schema.

Example: any pseudo-finite theory is a limit of stable theories. This includes a lot of of simple theories, e.g. the theory of the random graph. It also includes pseudo-finite linear orders, which are SOP.

Example: suppose $T$ is a completion of ACFA (the model companion of algebraically closed fields with an automorphism), so $T$ is simple. Suppose $\phi \in T$. Then $\phi$ is consistent with ACFA, so for arbitrarily large powers of primes $p^n$, $(F, x \mapsto x^{p^n}) \models \phi$, where $F$ is some (any) algebraically closed field of characteristic $p$. But these structures are stable.

A strong counterexample would be a finitely axiomatizable simple unstable theory.

Answer 1

No, not every simple theory is a limit of stable theories. For example, let $K$ be a pseudoalgebraically closed field with a small, nontrivial Galois group not isomorphic to $\widehat{\mathbb{Z}}$. Possibly after replacing $K$ with a finite algebraic extension, there will be some natural number $n$ coprime to the characteristic for which the $n^\text{th}$ power map is not onto. By smallness of the Galois group, the group $K^\times/(K^\times)^n$ is finite, say of size $m$. That the absolute Galois group is not isomorphic to $\widehat{\mathbb{Z}}$ is reflected by there being some natural number $\ell$ for which either there is no extension of degree $\ell$ or there are two distinct extensions of degree $\ell$. The theory of $K$ is supersimple, but the sentence $\phi$ which asserts that

• $K$ is a field,
• that there is some $x \in K$ which is not an $n^\text{th}$ power,
• that there are $m$ elements $y_1, \ldots, y_m$ of $K$ so that every nonzero element of $K$ may be expressed as $x_i z^n$ for some $z \in K$ and $1 \leq i \leq m$, and
• that either there are no extensions of degree $\ell$ or there are two distinct such extensions

holds in $K$ but cannot hold in any stable structure: any such structure is a field (by the first item) and finite fields cannot satisfy the fourth item while infinite stable fields have connected multiplicative groups and hence cannot satisfy the conjunction of the second and third items.