Filter Logics on ω

Abstract

AbstractLogics LF(M) are considered, in which M (“most”) is a new first-order quantifier whose interpretation depends on a given filter F of subsets of ω. It is proved that countable compactness and axiomatizability are each equivalent to the assertion that F is not of the form {(⋂F) ∪ X: ∣ω − X∣ < ω} with ∣ω − ⋂F∣ = ω. Moreover the set of validities of LF (M) and even of depends only on a few basic properties of F. Similar characterizations are given of the class of filters F for which LF (M) has the interpolation or Robinson properties. An omitting types theorem is also proved. These results sharpen the corresponding known theorems on weak models (, where the collection q is allowed to vary. In addition they provide extensions of first-order logic which possess some nice properties, thus escaping from contradicting Lindström's Theorem [1969] only because satisfaction is not isomorphism-invariant (as it is tied to the filter F). However, Lindström's argument is applied to characterize the invariant sentences as just those of first-order logic.