Ideal models and some not so ideal problems in the model theory of L(Q)

Abstract

It is the purpose of this paper to investigate the model theory of logic with a generalized quantifier; in particular the logic L(Q1) where Q1xφ(x) has the intended meaning “there exist uncountably many x such that φ(x)”. We do this from the point of view that the best way to study what happens in the so-called “ω1-standard” models of L(Q1) is to examine the countable ideal models of L(Q) that satisfy all of the axioms for L(Q1) (see definitions of ω1-standard and ideal models in §1). We believe that this study can be as fruitful for L(Q1) as the study of countable models of ZF has been for set theory.A major problem is formulating an adequate definition of submodel for countable ideal models that is compatible with that for ω1-standard models. Thus we begin the paper by discussing several possible definitions of the notion of submodel. We then adopt a particular definition of submodel and investigate model-completeness in L(Q). We define model-completeness both for ω1-standard models and for countable ideal models and compare the two notions. We also examine elimination of quantifiers, as well as investigating formulas preserved under submodels, again both for ω1-standar d and countable ideal models.