The spectrum problem III: Universal theories

We solve the classification problem and essentially the spectrum problem for universal theories (see [6] for discussion of the meaning of this). We first solve it for T such that if M ₁, M ₂ elementarily extend M ₀ and are independent over it, then over M ₀ ∪M ₁ there is a prime model. This generalizes [2]. This was subsequently used and generalized for countable first order theories. (This will appear in [5].) But note that there the theory is countable and in the case of structure the model is prime over a non-forking tree of models; here the model is generated by the union (and the T not necessarily countable). The universality is used in Theorey. If T is stable and complete then either (A)for every M ₁<M (l=0, 1, 2)models of T, if M ₀ ⊆M ₁, M ₂, {M ₁, M ₂}is independent over M ₀ (i.e. tp(M ₁, M ₂)is finitely satisfiable in M ₀), then the submodel of M which M ₁ {n-ary union}M ₂ generates is an elementary submodel of M, or (B)there is an unstable theory extending the universal part of T (we can replace universal by Σ₂ and slightly more). Conclusion. For any universal T:Either (a) for every model M of T there is a tree I with ≦ω levels and submodels N _η (η ∈I) of power ≦2^|T| (by [5], just ≦|T|) such that (i)M is generated by ∪_ηεl N _η, (ii)η <v⇒⇒N _η, (iii) if v is an immediate successor of η then tp(N _v, {n-ary union}{{N _p:ρ ∈I, v{less-than but not equal to}ρ{variant}}) is finitely satisfiable in N _η (note that asking this just for quantifier-free formulas is enough). Or (b) for every cardinal λ>|T|, there are 2^γ non-isomorphic models for power λ.