Stable Theory

A Complete First-Order Theory is $\kappa$ stable for an infinite Cardinal $\kappa$ if for every set $A$ of cardinality $\kappa$ in a model $T$, the set $S(A)$ of complete types over $A$ also has cardinality $\kappa$.

An alternative definition is that stable theories do not have the order property. A theory $T$ has the order property if there is a formula $\phi(\bar{x},\bar{y})$ and two infinite sequences of typles $\bar{a}_{i},\bar{b}_{j}$ in a Model $M$ such that $\phi(\bar{a_{i}},\bar{b_{j}})$ is true in $M$ iff $i\leq j$.

We can also think about this in terms of Stone Spaces: a theory $T$ is $\kappa$-stable if the Stone Space if every $\kappa$-small Model of $T$ is also $\kappa$-small.