ω -stability

Abstract

We continue here our treatment of Morley’s Theorem and related ideas. All throughout this section, T is a complete theory with no finite models in a countable language L, and ωdenotes a big saturated model of T. In the last chapter we defined Morley rank and Morley degree as a complexity measure for definable sets. In particular we studied the simplest infinite definable sets with respect to this measure, i. e. the strongly minimal sets, those having Morley rank 1 and Morley degree 1; we considered also strongly minimal theories, i. e. the complete theories whose universe is strongly minimal. More generally, one can examine the theories in which every non empty definable set gets a(n ordinal) Morley rank. These were called by Morley totally transcendental.