Σ1-compactness and ultraproducts

Abstract

This paper is devoted to a description of the way in which ultraproducts can be used in proofs of various well-known Σ1-compactness theorems for infinitary languages ℒA associated with admissible sets A; the method generalises the ultra-product proof of compactness for finitary languages.The compactness theorems we consider are (§2) the Barwise Compactness Theorem for ℒA when A is countable admissible [1], and (§3) the Cofinality (ω) Compactness Theorem of Barwise and Karp [2] and [4]. Our proof of the Barwise theorem unfortunately has the defect that it relies heavily on the Completeness Theorem for ℒA. This defect has, however, been avoided in the case of the Cf(ω) Compactness Theorem, so we have a purely model-theoretic proof.