Filter Logics on omega

Logics L F ( 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 ∣ F ∣ = ω. Moreover the set of validities of L F ( 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 L F ( 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 Lindstrom's Theorem [1969] only because satisfaction is not isomorphism-invariant (as it is tied to the filter F ). However, Lindstrom's argument is applied to characterize the invariant sentences as just those of first-order logic.