A note on noncompact logics

TL;DR

This paper identifies conditions under which certain logics extending first-order logic fail to be compact, especially when expressing properties related to automorphism groups, using techniques from random graph theory.

Contribution

It introduces conditions linking properties of automorphism groups to the failure of compactness in extended logics, with new constructions using random graph theory.

Findings

Logics expressing certain automorphism properties lack compactness.

Construction of theories with rich automorphism groups but finite models are rigid.

Conditions under which extended logics fail to have the compactness property.

Abstract

A condition, in two variants, is given such that if a property P satisfies this condition, then every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The result is used to prove that for a number of natural properties P speaking about automorphism groups, every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The basic idea underlying the results and examples presented here is that, using results from random graph theory, it is possible to construct a countable first-order theory T such that every model of T has a very rich automorphism group, but every finite subset of T has a model which is rigid.