On compactness of logics that can express properties of symmetry or connectivity

TL;DR

The paper establishes conditions under which logics expressing certain symmetry or connectivity properties fail to be compact, highlighting limitations of strong logics in capturing these properties.

Contribution

It introduces conditions that determine when logics expressing symmetry or connectivity properties cannot be compact, and provides examples of such logics.

Findings

Logics expressing properties about automorphism groups or connectivity are non-compact.

An example of a compact logic expressing a bound on automorphism group size is provided.

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 or connectivity, every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. We also give an example of a logic that extends first-order logic, has the compactness property and can express the property "the cardinality of the automorphism group is at most $2^{\aleph_0}$".