THE STABILITY SPECTRUM FOR CLASSES OF ATOMIC MODELS

Abstract

We prove two results on the stability spectrum for Lω1,ω. Here [Formula: see text] denotes an appropriate notion (at or mod) of Stone space of m-types over M. (1) Theorem for unstable case: Suppose that for some positive integer m and for every α < δ(T), there is an M ∈ K with [Formula: see text]. Then for every λ ≥ |T|, there is an M with [Formula: see text]. (2) Theorem for strictly stable case: Suppose that for every α < δ (T), there is Mα ∈ K such that λα = |Mα| ≥ ℶα and [Formula: see text]. Then for any μ with μℵ0 > μ, K is not i-stable in μ. These results provide a new kind of sufficient condition for the unstable case and shed some light on the spectrum of strictly stable theories in this context. The methods avoid the use of compactness in the theory under study. In this paper, we expound the construction of tree indiscernibles for sentences of Lω1,ω. Further we provide some context for a number of variants on the Ehrenfeucht–Mostowski construction.