Models of Long Sentences I

Let L be a countable first order language. Let A be a Σ1 admissible set such that L ∈ A and the cardinality of A is ω1. Let T ⊆ LA,ω be a set of sentences such that < A, T > is Σ1 admissible. T is consistent iff no contradiction can be derived from T via a deduction in A. T is complete iff for each sentence F ∈ LA,ω, either F ∈ T or (¬F) ∈ T . An n-type p of T is a consistent, complete set of formulas of arity ≤ n such that < A, p > is Σ1 admissible; ST is the set of all types of T . Note that sentences of uncountable length may belong to T . Say T is degenerate iff T has a countable, ω-homogeneous model that realizes every type in ST . A typical instance of type-completeness is: if ∃yF(x, y) ∈ p(x) ∈ ST , then there is a q(x, y) ∈ ST such that p(x) ⊆ q(x, y) and F(x, y) ∈ q(x, y). T is type-admissible iff < A, p > is Σ1 admissible for each finite, coherent pair p of types. Main Results: If T is consistent, complete, type-complete, type-admissible, and not degenerate, then T has a model of cardinality ω1; Mild stability implies type-completeness and typeadmissibilty.