---
tag: C44JJKGCBF
kind: definition
name: Stable theory
msc: [03C45]
defines: T7JW5QY6KR
uses: [655K8Y7VS8, 9V71Q3SD1G, MVCH6N9SNW, 0EBACQFAZW, 0G0ASKNA6Z, MT8TDVPBXB, 2W4S2H4VYA]
sources:
  - url: https://en.wikipedia.org/wiki/Stable_theory
---

## Definition

Let $T$ be a [[complete theory|655K8Y7VS8]] in a countable signature, with infinite models, and let $\lambda$ be an infinite [[cardinal|0EBACQFAZW]].
The theory $T$ is *$\lambda$-stable* if the following holds for every [[model|MVCH6N9SNW]] $M$ of $T$ and every set $A$ of elements of $M$ with $\abs{A} \le \lambda$.
The set $S_1^M(A)$ of complete $1$-[[types|9V71Q3SD1G]] over $A$ has [[cardinality|0G0ASKNA6Z]] at most $\lambda$.
The theory $T$ is *stable* if it is $\lambda$-stable for some infinite cardinal $\lambda$.

## Remarks

An $\aleph_0$-stable theory is called $\omega$-stable.
Shelah proved that $T$ is stable if and only if no formula $\varphi(\bar{x}, \bar{y})$ has the order property.
The order property means that some model of $T$ has tuples $\bar{a}_i, \bar{b}_i$ for $i \in \N$ with $\varphi(\bar{a}_i, \bar{b}_j)$ true exactly when $i < j$.

Example: the theory $\mathrm{ACF}_0$ of [[algebraically closed fields|MT8TDVPBXB]] of characteristic $0$ is $\omega$-stable.
For a countable set $A$ in a model $K$, let $k$ be the subfield generated by $A$.
By quantifier elimination, a complete $1$-type over $A$ is determined by the ideal of polynomials in $k[x]$ that vanish at $x$.
This ideal is $0$ or is generated by an irreducible polynomial, and $k[x]$ has countably many polynomials, so there are countably many types.

Non-example: the theory of [[dense linear orders without endpoints|2W4S2H4VYA]] is not stable.
The formula $x < y$ has the order property: in $(\Q, <)$ take $a_i = b_i = i$, so $a_i < b_j$ exactly when $i < j$.
For $\lambda = \aleph_0$ one sees the failure directly, with $A = \Q$ inside the model $\R$.
Each irrational number $r$ realizes its own complete type over $\Q$, determined by the cut $\set{q \in \Q : q < r}$.
So $S_1^{\R}(\Q)$ has size at least $2^{\aleph_0}$.