Syntactic characterisations of amalgamation, convexity and related properties

Abstract

We prove that certain syntactic conditions similar to separation principles on a theory are equivalent to semantic properties such as amalgamation and strong amalgamation, by showing that appropriate classes of structures are definable by Lω1ω-sentences. Then we characterise the elements of core models and thus give a natural proof of Rabin's characterisation of convex theories.The notion of a syntactic characterisation of a semantic property of a theory is by now fairly well known. The earliest such were the classical preservation theorems: For example, a theorem of Lyndon characterised the theories whose models were closed under homomorphic images as those with a set of positive axioms.Presumably the notion of syntactic characterisation can be made precise, but it is probably better at this stage to leave it vague. The general idea is that theories are “algebras” (cylindric algebras, or logical categories, with suitable extra structure) and that a semantic property P of theories is syntactically characterisable if the class of theories with P is an “elementary” class of “algebras.”When one codes countable theories as real numbers, a syntactically characterisable property will be arithmetical. The converse does not seem reasonable, especially as it is often fairly easy to prove a property arithmetical (using extra predicates, usually), when we may not be able to find a syntactic characterisation.