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

Publisher： Cambridge University Press

E-ISSN： 1943-5886|43|2|304-321

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.43, Iss.2, 1978-06, pp. : 304-321

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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.

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