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The measure quantifier1

Published online by Cambridge University Press:  12 March 2014

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

Originally generalized quantifiers were introduced to specify that a given formula was true for “many x's” e.g. ⊨ Qxφ(x) iff card{x ∈ ∣∣ ∣ ⊨ φ[x]} ≥ ℵ0, ℵ1, or some fixed cardinal κ. In this paper we formalize the notion that φ{x) is true “for almost all x”. This is accomplished by referring to structures = (′, μ) where ′ is a first-order structure and μ is a measure of a suitable type on the universe of ′. We will prove that the language Lμ obtained from first-order logic by adjoining a quantifier Qμ, which ranges over the measure μ, is fully compact if we assume the existence of a proper class of measurable cardinals. As a corollary to the compactness theorem we obtain the recursive enumerability of the validities of Lμ. Finally, the Magidor-Malitz quantifiers Qkn (n ∈ ω) will be added to Lμ together with analogous quantifiers Qμm (m ∈ ω) to form Lκμ<ω,<ω, which is compact for sets of sentences of cardinality < κ, where κ is a measurable cardinal > ℵ0.

An alternate approach to formalizing “for almost all” has been recently developed by Barwise, Kaufmann and Makkai [1] who follow a suggestion of Shelah [5].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 44 , Issue 1 , March 1979 , pp. 103 - 108
Copyright
Copyright © Association for Symbolic Logic 1979

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Footnotes

1

This constitutes part of the author's Ph.D. thesis written at the University of Colorado under the supervision of Professor William N. Reinhardt to whom the author is grateful for his patience and guidance.

I am grateful to the referee for many helpful comments, in particular with regard to Theorem 4.1.

References

REFERENCES

[1]Barwise, J., Kaufmann, M. and Makkai, M., Stationary logic, Annals of Mathematical Logic, vol. 13(1978), pp. 171–224.CrossRefGoogle Scholar
[2]Drake, F., Set theory: An introduction to large cardinals, North-Holland, Amsterdam, 1974.Google Scholar
[3]Helling, M., Model theoretic problems for some extensions of first order languages, Ph.D. thesis, University of California, Berkeley, 1966.Google Scholar
[4]Magidor, M. and Malitz, J., Compact extensions of L(Q), Annals of Mathematical Logic, vol. 11(1977), pp. 217–261.CrossRefGoogle Scholar
[5]Shelah, S., Generalized quantifiers and compact logic, Transactions of the American Mathematical Society, vol. 24(1975), pp. 342–364.CrossRefGoogle Scholar
[6]Silver, J., Some applications of model theory in set theory, Ph.D. thesis, University of California, Berkeley, 1966; Annals of Mathematical Logic, vol. 3 (1971), pp. 45-110.Google Scholar
[7]Slomson, A.B., Some problems in mathematical logic, Ph.D. thesis, Oxford, 1967.Google Scholar