# Characterization of stable theories

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?

• What is $S^M_n(A)$? – Taroccoesbrocco Aug 12 '18 at 8:44
• $S_n^M(A)=\{ p : p$ is a complete type over $A \}$ – Lajos Aug 12 '18 at 13:49

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.
• Thank you very much Alex. Is there any characterization for the other unstable classes for instance NSOP1, NTP1, NSOP3 and etc in terms of non-dividing independence? Also thanks for introducing forkinganddividing.com to me. It's a nice website. – Lajos Aug 13 '18 at 15:14
• @Lajos These questions are at the forefront of current research. Kaplan and Ramsey recently made a breakthrough by showing that NSOP1 can be characterized, not by non-forking or non-dividing independence, but by a generalization called Kim independence. See their paper here: arxiv.org/abs/1702.03894 in particular Theorem 9.1. – Alex Kruckman Aug 13 '18 at 15:19
• Thanks a lot for your helpful comment. – Lajos Aug 13 '18 at 15:27
• Does the non-forking independence relation have a property that holds in strongly minimal theories but does not hold in stable theories? – Lajos Aug 14 '18 at 14:31
• @Lajos Well, in a strongly minimal theory, you can characterize forking in terms of algebraic dimension, and that characterization doesn't hold in an arbitrary stable theory. So in that sense yes. But to me that characterization has a different flavor than the usual kind of abstract properties like local character or stationarity. – Alex Kruckman Aug 14 '18 at 14:34