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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

REFERENCES

Barwise, J., Kaufmann, M. and Makkai, M. [1977], Stationary logic, Annals of Mathematical Logic (to appear).Google Scholar
Barwise, J. and Schlipf, J. [1976], An introduction to recursively saturated and resplendent models, this Journal, vol. 41, pp. 531–536.Google Scholar
Bell, J. L. and Slomson, A. B. [1969], Models and ultraproducts, North-Holland, Amsterdam.Google Scholar
Bruce, K. B. [1975], Model-theoretic forcing with a generalized quantifier, Ph.D. Thesis, University of Wisconsin.Google Scholar
Bruce, K. B. [1978], Model-theoretic forcing in logic with a generalized quantifier, Annals of Mathematical Logic (to appear).Google Scholar
Bruce, K. B. [1978a], Model constructions in stationary logic, Part I: Forcing (to appear).Google Scholar
Chang, C. C. and Keisler, H. J. [1973], Model theory, North-Holland, Amsterdam.Google Scholar
COWLES, J. [1975], Abstract logic and extensions of first-order logic, Ph.D. Thesis, Pennsylvania State University.Google Scholar
Feferman, Solomon [1974], Application of many-sorted interpolation theorems, Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics XXV, pp. 205–223, American Mathematical Society, Providence, R.I.Google Scholar
Friedman, H. [1973], Beth's Theorem in cardinality logics, Israel Journal of Mathematics, vol. 14, pp. 205–212.CrossRefGoogle Scholar
Hutchinson, J. [1976], Model theory via set theory, Israel Journal of Mathematics, vol. 24, pp. 286–304.CrossRefGoogle Scholar
Jech, T. [1973], Some combinatorial problems concerning uncountable cardinals, Annals of Mathematical Logic, vol. 5, pp. 165–198.CrossRefGoogle Scholar
Jensen, F. V. [1975], On completeness in cardinality logics, Bulletin de L'Academie Polonaise des Sciences, vol. 23, pp. 117–122.Google Scholar
Keisler, H. J. [1970], Logic with the quantifier ‘there exist uncountably many’, Annals of Mathematical Logic, vol. 1, pp. 1–93.CrossRefGoogle Scholar
Kueker, D. W. [1972], Lowenheim–Skolem and interpolation theorems in infinitary languages, Bulletin of the American Mathematical Society, vol. 78, pp. 211–215.CrossRefGoogle Scholar
Magidor, M. and Malitz, J. [1977], Compact extensions of L(Q), Part 1a, Annals of Mathematical cogic (to appear).Google Scholar
Robinson, A. [1974], Introduction to model theory and to the metamathematics of algebra, North-Holland, Amsterdam.Google Scholar
Schlipf, J. [1978], Toward model theory through recursive saturation, this Journal, vol. 43, pp. 183–206Google Scholar
Shelah, S. [1976], Personal communication.Google Scholar
Shelah, S. [1977], Personal communication.Google Scholar
Vinner, S. [1975], Model-completeness in a first-order language with a generalized quantifier, Pacific Journal of Mathematics, vol. 56, pp. 265–273.CrossRefGoogle Scholar