Stable theory

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 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 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 $2^{\aleph_0}$.