We prove that if T is a stable theory with only a finite number of countable models, then T contains a type-definable pseudoplane. We also show that for any stable theory T either T contains a type-definable pseudoplane or T is weakly normal.

We prove: Theorem. A complete first order theory in a countable language which is strictly stable, trivial and which admits finite coding has 2 ℵ 0 nonisomorphic countable models. Combined with the corresponding result or superstable theories from [4] our result confirms the Vaught conjecture for trivial theories which admit finite coding.

A stable theory is finitely based if every set of indiscernibles is based on a finite subset. This is a common generalization of superstability and 1-basedness. We show that if such theories have more than one model they must have infinitely many, and prove some other conjectures.

We define a generalized notion of rank for stable theories without dense forking chains, and use it to derive that every type is domination-equivalent to a finite product of regular types. We apply this to show that in a small theory admitting finite coding, no realisation of a nonforking extension of some strong type can be algebraic over some realisation of a forking extension.

We prove: Theorem. A complete first order theory in a countable language which is strictly stable, trivial and which admits finite coding has $2^{\aleph_0}$ nonisomorphic countable models. Combined with the corresponding result or superstable theories from [4] our result confirms the Vaught conjecture for trivial theories which admit finite coding.