Abstract Model Structures and Compactness Theorems

TL;DR

This paper introduces a generalized, logic-independent notion of compactness using abstract model structures, along with characterization theorems for specific classes of these structures.

Contribution

It develops a unified framework for compactness theorems that does not rely on particular logical syntactic or semantic features.

Findings

Generalized compactness concept independent of specific logics

Characterization theorems for certain classes of abstract model structures

Framework applicable across various logical systems

Abstract

The compactness theorem for a logic states, roughly, that the satisfiability of a set of well-formed formulas can be determined from the satisfiability of its finite subsets, and vice versa. Usually, proofs of this theorem depend on the syntactic/semantic particularities of the corresponding logic. In this paper, using the notion of \emph{abstract model structures}, we show that one can develop a generalized notion of compactness that is independent of these. Several characterization theorems for a particular class of compact abstract model structures are also proved.