Model constructions in stationary logic. Part I. Forcing

Abstract

In this series of two papers we wish to discuss two new methods of constructing models for an arbitrary theory T of stationary logic, L(aa). The original work in constructing models for this logic appears in Barwise-Kaufmann-Makkai [BKM] (which we shall refer to as BKM for brevity). In that paper the authors prove completeness, compactness, and omitting types theorems for L(aa) by a technique similar to that used by Keisler [1970] in constructing models of L(Q1), logic with the generalized quantifier “there exist uncountably many”. In this paper we give an alternative method of constructing models using a method of model-theoretic forcing similar to that of Bruce [1978]. In a succeeding paper we will give yet another method of constructing models using definable ultrapowers.As applications of this new forcing construction we give new proofs of the completeness and omitting types theorems for LB(aa), where LB(aa) is any countable fragment of Lω1ω(aa). We also give new proofs of versions of the Barwise completeness and compactness theorems for LA(aa) where A is a countable admissible set. In proofs of these theorems the model constructions via forcing are used in a way very similar to that of consistency properties in Lω1ω.Aside from the ease in handling the model theory of Lω1ω(aa) we see two other advantages in this sort of model construction. The first is in the ability to prove versions of theorems relativized to formulas of low complexity (e.g. relativized omitting types theorems). The second is that we have a high degree of control over the construction of standard models and hence have a better knowledge of the structure of the model (see the remarks following the proof of Lemma 3.3), something that should prove useful in constructing more complicated models.