Abstract
We introduce a natural class of quantifiersTh containing all monadic type quantifiers, all quantifiers for linear orders, quantifiers for isomorphism, Ramsey type quantifiers, and plenty more, showing that no sublogic ofL ωω(Th) or countably compact regular sublogic ofL ∞ω(Th), properly extendingL ωω, satisfies the uniform reduction property for quotients. As a consequence, none of these logics satisfies eitherΔ-interpolation or Beth's definability theorem when closed under relativizations. We also show the failure of both properties for any sublogic ofL ∞ω(Th) in which Chang's quantifier or some cardinality quantifierQ α, with α≧1, is definable.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barwise, J., Feferman, S.: Model theoretic logics. Berlin Heidelberg New York: Springer 1985
Caicedo, X.: Failure of interpolation for quantifiers of monadic type. In: Methods in Mathematical Logic, Lect. Notes Math., vol. 1130, pp. 1–12. Berlin Heidelberg New York: Springer 1985
Caicedo, X.: A simple solution to Friedman's fourth problem. J. Symb. Logic51, 778–784 (1986)
Ebbinghaus, H.D.: Extended logics: the general framework, Chap. II. In: Barwise, J., Feferman, S. (eds.) Model theoretic logics. Berlin Heidelberg New York: Springer 1985
Feferman, S.: Two notes in abstract model theory. I. Fundam. Math.LXXXII, 153–156 (1974)
Friedman, H.: Beth's theorem in cardinality logics. Isr. J. Math.14, 205–213 (1973)
Hella, L.: Definability hierarchies of generalized quantifiers. Ann. Pure Appl. Logic43, 235–271 (1989)
Krynicki, M.: Notion of interpretation and non elementary languages. Z. Math. Logik Grundlagen Math. (to appear)
Krynicki, M., Lachlan, A., Väänänen, J.: Vector spaces and binary quantifiers. Notre Dame J. Formal Logic25, 72–78 (1984)
Makowsky, J.A.: Compactness, embeddings and definability, Chap. XVIII. In: Barwise, J., Feferman, S. (eds.) Model theoretic logics. Berlin Heidelberg New York: Springer 1985
Makowsky, J.A., Shelah, S.: The theorems of Beth and Craig in abstract model theory. I. Trans. Am. Math. Soc.256, 215–239 (1979)
Makowsky, J.A., Shelah, S.: The theorems of Beth and Craig in abstract model theory. II. Archiv. Math. Logik21, 13–35 (1981)
Mekler, A.H., Shelah, S.: Stationary logic and its friends. I. Notre Dame J. Formal Logic26, 129–138 (1985)
Mekler, A.H., Shelah, S.: Stationary logic and its friends. II. Notre Dame J. Formal Logic27, 39–50 (1986)
Mundici, D.: Other quantifiers: an overview, Chap. VI. In: Barwise, J., Feferman, S. (eds.) Model theoretic logics. Berlin Heidelberg New York: Springer 1985
Shelah, S.: Generalized quantifiers and compact logics, Trans. Am. Math. Soc.204, 342–364 (1975)
Shelah, S.: Remarks on abstract model theory. Ann. Pure Appl. Logic29, 255–288 (1985)
Väänänen, J.: Remarks on generalized quantifiers and second order logics. Pr. Nauk. Inst. Mat. Politech. Wroclaw14, 117–123 (1977)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Caicedo, X. Definability properties and the congruence closure. Arch Math Logic 30, 231–240 (1990). https://doi.org/10.1007/BF01792985
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01792985