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

Published online by Cambridge University Press:  12 March 2014

Kim B. Bruce
Kim B. Bruce*
Affiliation:
Williams College, Williamstown, MA 01267
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Extract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 2 , June 1978 , pp. 304 - 321
Copyright
Copyright © Association for Symbolic Logic 1978

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References

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