Israel Journal of Mathematics

, Volume 47, Issue 2–3, pp 131–138 | Cite as

Separating ultrafilters on uncountable cardinals

  • Aki Kanamori
  • Alan D. Taylor


A uniform ultrafilterU on κ is said to be λ-separating if distinct elements of the ultrapower never projectU to the same uniform ultrafilterV on λ. It is shown that, in the presence of CH, an ω-separating ultrafilterU on κ>ω is non-(ω, ω1)-regular and, in fact, if κ < ℵω thenU is λ-separating for all λ. Several large cardinal consequences of the existence of such an ultrafilterU are derived.


Distinct Element Regular Cardinal Measurable Cardinal Easy Exercise Uncountable Cardinal 
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  1. 1.
    M. Benda and J. Ketonen,Regularity of ultrafilters, Isr. J. Math.17 (1974), 231–240.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. Blass,Orderings of ultrafilters, Thesis, Harvard, 1970.Google Scholar
  3. 3.
    C. C. Chang,Descendingly incomplete ultrafilters, Trans. Am. Math. Soc.126 (1967), 108–118.MATHCrossRefGoogle Scholar
  4. 4.
    G. Chudnovsky and D. Chudnovsky,Regularnye i ubyvajusche nepolnye ultrafiltry, Dokl. Akad. Nauk SSR198 (1971), 779–782.Google Scholar
  5. 5.
    H. Donder, R. Jensen and B. Koppelberg,Some applications of the core model, inSet Theory and Model Theory (R. B. Jensen and A. Prestel, eds.), Lecture Notes in Math., No. 872, Springer-Verlag, 1981, pp. 55–97.Google Scholar
  6. 6.
    M. Jorgensen,Images of ultrafilters and cardinality of ultrapowers, Am. Math. Soc. Notices18 (1971), 826.Google Scholar
  7. 7.
    A. Kanamori,Weakly normal filters and irregular ultrafilters, Trans. Am. Math. Soc.220 (1976), 393–399.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. Ketonen,Some combinatorial properties of ultrafilters, Fund. Math.107 (1980), 225–235.MATHMathSciNetGoogle Scholar
  9. 9.
    K. Kunen and K. Prikry,On descendingly incomplete ultrafilters, J. Symb. Logic36 (1971), 650–652.CrossRefMathSciNetGoogle Scholar
  10. 10.
    A. Mekler, D. Pelletier and A. Taylor,A note on a lemma of Shelah concerning stationary sets, Proc. Am. Math. Soc.83 (1981), 764–768.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    D. Pelletier,A result on the cardinality of ultrapowers, Abstracts Am. Math. Soc. 82T-04-144,3 (1982), 185.Google Scholar
  12. 12.
    K. Prikry,On descendingly complete ultrafilters, inCambridge Summer School in Mathematical Logic (A. R. D. Mathias and H. Rogers Jr., eds.), Lecture Notes in Math., No. 337, Springer-Verlag, 1973, pp. 459–488.Google Scholar
  13. 13.
    A. Taylor,Regularity properties of ideals and ultrafilters, Ann. Math. Logic16 (1979), 33–55.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1984

Authors and Affiliations

  • Aki Kanamori
    • 1
    • 2
  • Alan D. Taylor
    • 1
    • 2
  1. 1.Boston UniversityBostonUSA
  2. 2.Union CollegeSchenectadyUSA

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