A note on noncompact logics

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.