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On amalgamations of languages with Magidor-Malitz quantifiers

Published online by Cambridge University Press:  12 March 2014

Carl F. Morgenstern
Carl F. Morgenstern*
Affiliation:
University of California, Santa Cruz, California 95064
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Extract

In this paper we indicate how compact languages containing the Magidor-Malitz quantifiers Qκn in different cardinalities can be amalgamated to yield more expressive, compact languages.

The language Lκ<ω, originally introduced by Magidor and Malitz [9], is a natural extension of the language L(Q) introduced by Mostowski and investigated by Fuhrken [6], [7], Keisler [8] and Vaught [13]. Intuitively, Lκ<ω is first-order logic together with quantifiers Qκn (n ∈ ω) binding n free variables which express “there is a set X of cardinality κ such than any n distinct elements of X satisfy …”, or in other words, iff the relation on determined by φ contains an n-cube of cardinality κ. With these languages one can express a variety of combinatorial statements of the type considered by Erdös and his colleagues, as well as concepts in universal algebra which are beyond the scope of first-order logic. The model theory of Lκ<ω has been further developed by Badger [1], Magidor and Malitz [10] and Shelah [12].

We refer to a language as being < κ compact if, given any set of sentences Σ of the language, if Σ is finitely satisfiable and ∣Σ∣ < κ, then Σ has a model. The phrase countably compact is used in place of <ℵ1 compact.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 44 , Issue 4 , December 1979 , pp. 549 - 558
Copyright
Copyright © Association for Symbolic Logic 1979

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References

REFERENCES

[1]Badger, L., The Malitz quantifier meets its Ehrenfeucht game, Ph.D. thesis, University of Colorado, 1975.Google Scholar
[2]Burgess, J., unpublished manuscript.Google Scholar
[3]Chang, C. C. and Keisler, J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[4]Drake, F., Set theory: An introduction to large cardinals, North-Holland, Amsterdam, 1974.Google Scholar
[5]Dushnik, B. and Miller, E. W., Partially ordered sets, American Journal of Mathematics, vol. 63(1941), pp. 600–610.CrossRefGoogle Scholar
[6]Fuhrken, G., Skolem-type normal forms for first order languages with a generalized quantifier, Fundamenta Mathematicae, vol. 54(1964), pp. 291–302.CrossRefGoogle Scholar
[7]Fuhrken, G., Languages with added quantifier “there exists at least ℵα”, The theory of models (Addison, J. W., Henkin, L., Tarski, A., Editors), North-Holland, Amsterdam, 1965, pp. 121–131.Google Scholar
[8]Keisler, J., Logic with the quantifier “there exists uncoimtably many”, Annals of Mathematical Logic, vol. 1(1970), pp. 1–93.CrossRefGoogle Scholar
[9]Magidor, M. and Malitz, J., Compact extensions of L(Q), Part 1a, Annals of Mathematical Logic, vol. 11(1977), pp. 217–261.CrossRefGoogle Scholar
[10]Magidor, M. and Malitz, J., Compactness and transfer for a fragment of L<ω, this Journal, vol. 42(1977), pp. 261–269.Google Scholar
[11]Mostowski, A., On a generalization of quantifiers, Fundamenta Mathematicae, vol. 44(1957), pp. 12–36.CrossRefGoogle Scholar
[12]Shelah, S., Various results in mathematical logic, Notices of the American Mathematical Society, vol. 24(1977), p. A–250.Google Scholar
[13]Vaught, R., The completeness of logic with the added quantifier “there are uncoimtably many”, Fundamenta Mathematicae, vol. 54(1964), pp. 303–304.CrossRefGoogle Scholar
[14]Vaught, R., A Löwenheim-Skolem theorem for cardinals far apart, The theory of models (Addison, J. W., Henkin, L., Tarski, A., Editors), North-Holland, Amsterdam, 1965, pp. 390–401.Google Scholar