Archive for Mathematical Logic

, Volume 32, Issue 6, pp 429–442 | Cite as

Stationary Cardinals

  • Wenzhi Sun


This paper will define a new cardinal called aStationary Cardinal. We will show that every weakly11-indescribable cardinal is a stationary cardinal, every stationary cardinal is a greatly Mahlo cardinal and every stationary set of a stationary cardinal reflects. On the other hand, the existence of such a cardinal is independent of that of a11-indescribable cardinal and the existence of a cardinal such that every stationary set reflects is also independent of that of a stationary cardinal. As applications, we will show thatV=L implies ◊κ1 holds if κ is11-indescribable and so forth.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baumgartner, J.: Ineffability properties of cardinals. I. In: Infinite and finite Sets. Colloq. Math. Soc. Janos Bolyai 10, pp. 109–130. Kesthely, Hungary, 1973Google Scholar
  2. 2.
    Baumgartner, J., Taylor, A., Wagon, S.: On splitting stationary subsets of large cardinals. J. Symb. Logic42, 203–214 (1977)Google Scholar
  3. 3.
    Baumgartner, J.: A new Class of Order Type. Ann. Math. Logic9, 187–222 (1976)Google Scholar
  4. 4.
    Boos, W.: Boolean extensions which efface the Mahlo properties. J. Symb. Logic39, 254–268 (1974)Google Scholar
  5. 5.
    Gitik, M., Shelah, S.: Cardinal preserving ideals (to appear)Google Scholar
  6. 6.
    Harrington, L., Shelah, S.: Some exact equiconsistency results in set theory. Notre Dame J. Formal Logic26, 178–188 (1985)Google Scholar
  7. 7.
    Jech, T.: Multiple forcing. Cambridge: Cambridge Press 1986Google Scholar
  8. 8.
    Jech, T., Prikry, K.: Ideals over uncountable sets: application of almost disjoint functions and generic ultrapowers. Mem. Am. Math.18, 214 (1979)Google Scholar
  9. 9.
    Jensen, R.: The fine structure of the constructible hierarchy. Ann. Math. Logic4, 229–308 (1972)Google Scholar
  10. 10.
    Johnson, C.: On ideals and stationary reflections. J. Symb. Logic54, 568–575 (1989)Google Scholar
  11. 11.
    Kunen, K.: Boolean and measurable cardinals. Ann. Math. Logic2, 359–377 (1971)Google Scholar
  12. 12.
    Leary, C.: Superstationary cardinal and ineffable cardinals. Arch. Math. Logic29, 137–148 (1990) (3)Google Scholar
  13. 13.
    Magidor, M.: Reflecting stationary sets. J. Symb. Logic47, 755–771 (1982)Google Scholar
  14. 14.
    Mekler, A.: The consistency strength of Every stationary set reflects. Isr. J. Math.67, 353–366 (1989) (3)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Wenzhi Sun
    • 1
  1. 1.Salem CollegeWinston-SalemUSA

Personalised recommendations