Characterization of stable theories

Question

Let $\kappa$ be an infinite cardinal. A theory $T$ is called $\kappa$-stable if for all model $M\models T$ and all $A\subset M$ with $|A|\leq \kappa$ we have $|S_n^M(A)|\leq \kappa$.

A theory $T$ is called stable if it is $\kappa$-stable for some infinite cardinal $\kappa$.

Question. Is there any characterization for stable theories in terms of non-forking independence relation similar to Kim-Pillay characterization for simple theories?

Answer 1

Yes. The easy way to see this uses the characterization of stable theories as those simple theories for which non-forking independence satisifies stationarity over models.

Stationarity: For any model $M$ and any set $B$, if $a$ is independent from $B$ over $M$, $a'$ is independent from $B$ over $M$, and $\text{tp}(a/M) = \text{tp}(a'/M)$, then $\text{tp}(a/MB) = \text{tp}(a'/MB)$.

In other words (together with the extension property), any type over $M$ has a unique independent extension to a type over $MB$.

So you can obviously characterize non-forking independence in stable theories by just taking the Kim-Pillay characterization of non-forking independence in simple theories and adding stationarity over models.

Actually, you can get away with fewer axioms than this; in particular, stationarity over models gives you the independence theorem over models for free. So there are various lists of axioms characterizing non-forking in stable theories:

• You can find a stable variant of the Kim-Pillay axioms in Casanovas's book Simple Theories and Hyperimaginaries, Definition 12.2 and Theorem 12.22.
• The fact that non-forking independence in stable theories can be axiomatically characterized may have first been noticed by Harnik and Harrington (long before Kim and Pillay started studying simple theories), see Axioms 0-3 and Theorem 5.8 in their paper Fundamentals of Forking.
• A more modern presentation of the Harnik and Harrington theorem is Theorem 8.5.10 in Tent and Ziegler's book A Course in Model Theory.