On Cantor-Bendixson spectra containing (1,1). II

Extract

Let T be a (countable, complete, quantifier eliminable) ω-stable theory; an analysis of T, and consequently a classification of ω-stable theories, can be done by looking at the Boolean algebras B(M) of definable subsets of its countable models M (as usual, we often confuse a definable subset of M with the class of formulas defining it). If M ⊨ T, ∣M∣ = ℵ₀, then, for every L_M-formula ϕ(v) and for every ordinal α, we define a relation

(CB = Cantor-Bendixson, of course) by induction on α:

CB-rank ϕ(v) ≥ 0 if ϕ(M) ≠ ∅

CB-rank ϕ(v) ≥ λ for λ a limit ordinal, if CB-rank ϕ(v) ≥ for all v < λ;

CB-rank ϕ(v)≥ α + 1 if, for all n ∈ ω,

(*) there are L_M-formulas ϕ₀(v), …, ϕ_{n − 1}(v) such that

It is well known that the ω-stability of T implies that, for every consistent L_M-formula ϕ(v), there is exactly one ordinal α < ω₁ such that CB-rank ϕ(v) ≥ α and CB-rank ϕ(v)≱α + 1. Therefore we define:

CB-rank ϕ(v) = α

CB-degree ϕ(v) = d if d is the maximal n ∈ ω satisfying (*); and

CB-type ϕ(v) = (α, d).