Abstract
We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assumingV=L, Gödel's Axiom of Constructibility, we prove that if λ > ωα then the logic with the quantifier “there existα many” is (λ,λ)-compact if and only if either λ is weakly compact or λ is singular of cofinality<ωα. As a corollary, for every infinite cardinals λ and μ, there exists a (λ,λ)-compact non-(μ,μ)-compact logic if and only if either λ<μ orcfλ<cfμ or λ<μ is weakly compact.
Counterexamples are given showing that the above statements may fail, ifV=L is not assumed.
However, without special assumptions, analogous results are obtained for the stronger notion of [λ,λ]-compactness.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[AH] Apter, A., Henle, J.: Relative consistency results via strong compactness. Fund. Math.139, 133–149 (1991)
[AJ] Adler, A., Jorgensen, M.: Descendingly incomplete ultrafilters and the cardinality of ultrapowers. Canad. J. Math.24, 830–834 (1972)
[BF] Barwise, J., Feferman, S. (eds.): Model-theoretic logics. In: Perspectives in Mathematical Logic, vol. 6. Berlin Heidelberg New York: Springer 1985
[BK] Benda, M., Ketonen, J.: Regularity of ultrafilters. Israel J. Math.17, 231–240 (1974)
[BS] Bell, J.L., Slomson, A.B.: Models and ultraproducts: an introduction. Amsterdam: North Holland 1969
[BM] Ben-David, S., Magidor, M.: The weak □* is really weaker than the full □. J. Symb. Logic51, 1029–1033 (1986)
[Bo] Boos, W.: Infinitary compactness without strong inaccessibility. J. Symb. Logic41, 33–38 (1976)
[CC] Cudnovskii, G.V., Cudnovskii, D.V.: Regular and descending incomplete ultrafilters (English translation). Soviet Math. Dokl.12, 901–905 (1971)
[CK] Chang, C.C., Keisler, H.J.: Model Theory, 3rd edn. Amsterdam: North Holland 1990
[CN] Comfort, W., Negrepontis, S.: The Theory of Ultrafilters. Berlin Heidelberg New York: Springer 1974
[Do] Donder, H.-D.: Regularity of ultrafilters and the core model. Israel J. Math.63, 289–322 (1988)
[KM] Kanamori, A., Magidor, M.: The evolution of large cardinal axioms in Set Theory. In: Higher Set Theory. Lecture Notes in Mathematics 669. Berlin Heidelberg New York: Springer 1978
[Ke] Keisler, J.: On cardinalities of ultraproducts. Bull. A.M.S.70, 644–647 (1964)
[Lp1] Lipparini, P.: Compactness of cardinality logics and constructibility. Proceedings of the 10th Easter conference on Model Theory. Seminarberichte of Humboldt Universität, No. 93-1, pp. 130–133, Berlin (1993)
[Lp2] Lipparini, P.: Limit ultrapowers and abstract logics. J. Symb. Logic52, 437–454 (1987)
[Lp3] Lipparini, P.: The compactness spectrum of abstract logics, large cardinals and combinatorial principles. Boll. U.M.I., Ser. VII, Vol.4-B, 875–903 (1990)
[Lp4] Lipparini, P.: About some generalizations of (λ,μ)-compactness. In: Proceedings of the Fifth Easter Conference in Model Theory. Seminarberichte of Humboldt Universität, No. 93, pp. 139–141, Berlin (1987)
[Lp5] Lipparini, P.: Consequences of compactness properties for abstract logics. Atti della Accademia Nazionale dei Lincei, Rendicotti Classe di Scienze fisiche, matematiche e naturali, vol. LXXX, pp. 501–503 (1986)
[Ma] Magidor, M.: On the singular cardinals problem I. Israel J. Math.28, 1–31 (1977)
[MR] Malitz, J., Reinhardt, W.N.: Maximal models in the language with quantifier “there exist uncountably many”. Pacific J. Math.40, 139–155 (1972)
[Pr] Prikry, K.L.: On descendingly complete ultrafilters. In: Mathias, A.R.D., Rogers, H. (eds.) Cambridge Summer School in Mathematical Logic. LNM337, 459–488 (1973)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lipparini, P. Ultrafilter translations. Arch Math Logic 35, 63–87 (1996). https://doi.org/10.1007/BF01273686
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01273686