Journal of the Mathematical Society of Japan

(κ,θ)-weak normality

Shimon GARTI and Saharon SHELAH

Full-text: Open access


We deal with the property of weak normality (for non-principal ultrafilters). We characterize the situation of | i<κ λ i /D|=λ . We have an application for a question of Depth in Boolean algebras.

Article information

J. Math. Soc. Japan, Volume 64, Number 2 (2012), 549-559.

First available in Project Euclid: 26 April 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E04: Ordered sets and their cofinalities; pcf theory
Secondary: 03G05: Boolean algebras [See also 06Exx]

set theory ultrafilters weak normality Boolean algebras Depth measurable cardinal


GARTI, Shimon; SHELAH, Saharon. (κ,θ)-weak normality. J. Math. Soc. Japan 64 (2012), no. 2, 549--559. doi:10.2969/jmsj/06420549.

Export citation


  • C. C. Chang and H. J. Keisler, Model theory, Stud. Logic Found. Math., 73, North-Holland Publishing Co., Amsterdam, 1973.
  • M. Foreman, M. Magidor and S. Shelah, Martin's maximum, saturated ideals and nonregular ultrafilters, II, Ann. of Math. (2), 127 (1988), 521–545.
  • S. Garti and S. Shelah, Depth of Boolean algebras, Notre Dame J. Form. Log., 52 (2011), 307–314.
  • T. Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978.
  • A. Kanamori, The higher infinite, Large cardinals in set theory from their beginnings, Perspect. Math. Logic, Springer-Verlag, Berlin, 1994.
  • J. König, Über die Grundlagen der Mengenlehre und das Kontinuumproblem, Math. Ann., 61 (1905), 156–160.
  • M. Magidor, How large is the first strongly compact cardinal? or A study on identity crises, Ann. Math. Logic, 10 (1976), 33–57.
  • J. D. Monk, Cardinal invariants on Boolean algebras, Progr. Math., 142, Birkhäuser Verlag, Basel, 1996.
  • S. Shelah, On the cardinality of ultraproduct of finite sets, J. Symbolic Logic, 35 (1970), 83–84.
  • S. Shelah, Classification theory and the number of nonisomorphic models, Stud. Logic Found. Math., 92, North-Holland Publishing Co., Amsterdam, 1978.
  • S. Shelah, Advances in cardinal arithmetic, Finite and infinite combinatorics in sets and logic, Banff, AB, 1991, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 411, Kluwer Acad. Publ., Dordrecht, 1993, pp.,355–383.
  • S. Shelah, The depth of ultraproducts of Boolean algebras, Algebra Universalis, 54 (2005), 91–96.
  • W. H. Woodin, The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter Ser. Log. Appl., 1, Walter de Gruyter & Co., Berlin, 1999.