Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T17:29:17.995Z Has data issue: false hasContentIssue false
  • English
  • Français

Some contributions to definability theory for languages with generalized quantifiers

Published online by Cambridge University Press:  12 March 2014

John T. Baldwin  and
Douglas E. Miller
John T. Baldwin
Affiliation:
University of Illinoisat Chicago Circle, Chicago, Illinois 60680
Douglas E. Miller
Affiliation:
University of Illinoisat Chicago Circle, Chicago, Illinois 60680
Get access Rights & Permissions [Opens in a new window]

Extract

One of the first results in model theory [12] asserts that a first-order sentence is preserved in extensions if and only if it is equivalent to an existential sentence.

In the first section of this paper, we analyze a natural program for extending this result to a class of languages extending first-order logic, notably including L(Q) and L(aa), respectively the languages with the quantifiers “there exist un-countably many” and “for almost all countable subsets”.

In the second section we answer a question of Bruce [3] by showing that this program cannot resolve the question for L(Q). We also consider whether the natural class of “generalized Σ-sentences” in L(Q) characterizes the class of sentences preserved in extensions, refuting the relativized version but leaving the unrestricted question open.

In the third section we show that the analogous class of L(aa)-sentences preserved in extensions does not include (up to elementary equivalence) all such sentences. This particular candidate class was nominated, rather tentatively, by Bruce [3].

In the fourth section we show that under rather general conditions, if L is a countably compact extension of first-order logic and T is an ℵ1-categorical first-order theory, then L is trivial relative to T.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 3 , September 1982 , pp. 572 - 586
Copyright
Copyright © Association for Symbolic Logic 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Baldwin, J. T. and Kueker, D. W., Ramsey quantifiers and the finite cover property, Pacific Journal of Mathematics, vol. 90 (1980), pp. 11–19.CrossRefGoogle Scholar
[2]Barwise, J., Kaufmann, M. and Makkai, M., Stationary logic, Annals of Mathematical Logic, vol. 13 (1978), pp. 171–224.CrossRefGoogle Scholar
[3]Bruce, K., Ideal models and some not so ideal problems in the model theory of L(Q), this Journal, vol. 43 (1978), pp. 304–321.Google Scholar
[4]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[5]Eklof, P. and Mekler, A., Stationary logic of finitely determinate structures, Annals of Mathematical Logic, vol. 17 (1979), pp. 227–271.CrossRefGoogle Scholar
[6]Fuhrken, G., Languages with added quantifier “there exist at least ℵα”, Theory of models (Addison, , Henkin, and Tarski, , Editors), North-Holland, Amsterdam, 1965, pp. 121–131.Google Scholar
[7]Keisler, H. J., Logic with the quantifier “there exist uncountably many”, Annals of Mathematical Logic, vol. 1 (1970), pp. 1–93.CrossRefGoogle Scholar
[8]Magidor, M. and Malitz, J., Compact extensions of L(Q), (part la), Annals of Mathematical Logic, vol. 11 (1977), pp. 217–263.CrossRefGoogle Scholar
[9]Makowsky, J. A. and Tulipiani, S., Some model theory for monotone quantifiers, Archir für Mathematische Logik und Grundlagen der Mathematik, vol. 13 (1975), pp. 115–134.Google Scholar
[10]Robinson, A., Note on a problem of L. Henkin, this Journal, vol. 21 (1956), pp. 33–35.Google Scholar
[11]Slomson, A., Decision problems for generalized quantifiers, Set Theory and Hierarchy Theory, Lecture Notes in Mathematics, vol. 537, Springer-Verlag, Berlin, 1976, pp. 249–258.Google Scholar
[12]Tarski, A., Contributions to the theory of models. I, II, Indagationes Mathematicae, vol. 16 (1954), pp. 572–588.CrossRefGoogle Scholar
[13]Vaught, R. L., The completeness of logic with the added quantifier, “there exist uncountably many”, Fundamenta Mathematicae, vol. 54 (1964), pp. 303–304.CrossRefGoogle Scholar