Journal of the Mathematical Society of Japan

Stationary reflection and the club filter

Masahiro SHIOYA

Full-text: Open access


Suppose that κ is a regular uncountable cardinal. It has been known that the club filter on P ω 1 κ can be presaturated. In this paper we extend the result to the case of P μ κ , where μ is a regular uncountable cardinal κ . This involves suitably weakening the notion of presaturation. A new reflection principle for stationary sets in P κ λ plays a key role.

Article information

J. Math. Soc. Japan, Volume 59, Number 4 (2007), 1045-1065.

First available in Project Euclid: 10 December 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E05: Other combinatorial set theory 03E35: Consistency and independence results 03E55: Large cardinals

large cardinal forcing presaturation


SHIOYA, Masahiro. Stationary reflection and the club filter. J. Math. Soc. Japan 59 (2007), no. 4, 1045--1065. doi:10.2969/jmsj/05941045.

Export citation


  • [1] \auJ. Baumgartner and A. Taylor, Saturation properties of ideals in generic extensions. II, \tiTrans. Amer. Math. Soc., , 271 ((1982),)\spg587–\epg609.
  • [2] M. Bekkali, Topics in Set Theory, Lecture Notes in Math., 1476, Springer, Berlin, 1991.
  • [3] \auD. Burke and Y. Matsubara, The extent of strength in the club filters, \tiIsrael J. Math., , 114 ((1999),)\spg253–\epg263.
  • [4] \auM. Foreman and M. Magidor, Large cardinals and definable counterexamples to the continuum hypothesis, \tiAnn. Pure Appl. Logic, , 76 ((1995),)\spg47–\epg97.
  • [5] \auM. Foreman, M. Magidor and S. Shelah, Martin's Maximum, saturated ideals, and non-regular ultrafilters. Part I, \tiAnn. Math., , 127 ((1988),)\spg1–\epg47.
  • [6] \auF. Galvin, T. Jech and M. Magidor, An ideal game, \tiJ. Symbolic Logic, , 43 ((1978),)\spg284–\epg292.
  • [7] \auM. Gitik, Some results on the nonstationary ideal, \tiIsrael J. Math., , 92 ((1995),)\spg61–\epg112.
  • [8] \auM. Gitik and S. Shelah, Forcings with ideals and simple forcing notions, \tiIsrael J. Math., , 68 ((1989),)\spg129–\epg160.
  • [9] \auM. Gitik and S. Shelah, Cardinal preserving ideals, \tiJ. Symbolic Logic, , 64 ((1999),)\spg1527–\epg1551.
  • [10] \auN. Goldring, Woodin cardinals and presaturated ideals, \tiAnn. Pure Appl. Logic, , 55 ((1992),)\spg285–\epg303.
  • [11] \auN. Goldring, The entire NS ideal on $\mathcal{P}_{\gamma}\mu$ can be precipitous, \tiJ. Symbolic Logic, , 62 ((1997),)\spg1161–\epg1172.
  • [12] T. Jech and K. Prikry, Ideals over uncountable sets: Application of almost disjoint functions and generic ultrapowers, Mem. Amer. Math. Soc., 18 (1979).
  • [13] A. Kanamori, The Higher Infinite, Springer Monogr. in Math., Springer, Berlin, 2003.
  • [14] P. Larson, The Stationary Tower, Notes on a Course by W. Hugh Woodin. Univ. Lecture Ser., 32, Amer. Math. Soc., Providence, RI, 2004.
  • [15] \auR. Laver, Making the supercompactness of $\kappa$ indestructible under $\kappa$-directed closed forcing, \tiIsrael J. Math., , 29 ((1978),)\spg385–\epg388.
  • [16] \auY. Matsubara, Stronger ideals over $\mathcal{P}_{\kappa}\lambda$, \tiFund. Math., , 174 ((2002),)\spg229–\epg238.
  • [17] \auY. Matsubara, Stationary preserving ideals over $\mathcal{P}_{\kappa}\lambda$, \tiJ. Math. Soc. Japan, , 55 ((2003),)\spg827–\epg835.
  • [18] \auY. Matsubara and S. Shelah, Nowhere precipitousness of the non-stationary ideal over $\mathcal{P}_{\kappa}\lambda$, \tiJ. Math. Logic, , 2 ((2002),)\spg81–\epg89.
  • [19] S. Shelah, Proper Forcing, Lecture Notes in Math., 940, Springer, Berlin, 1982.
  • [20] S. Shelah, Around Classification Theory of Models, Lecture Notes in Math., 1182, Springer, Berlin, 1986.
  • [21] S. Shelah, Nonstructure Theory, Oxford Univ. Press, to be published.
  • [22] \auS. Shelah and M. Shioya, Nonreflecting stationary sets in $\mathcal{P}_{\kappa}\lambda$, \tiAdv. Math., , 199 ((2006),)\spg185–\epg191.
  • [23] \auM. Shioya, The minimal normal $\mu$-complete filter on $P_{\kappa}\lambda$, \tiProc. Amer. Math. Soc., , 123 ((1995),)\spg1565–\epg1572.
  • [24] \auM. Shioya, Splitting $\mathcal{P}_{\kappa}\lambda$ into maximally many stationary sets, \tiIsrael J. Math., , 114 ((1999),)\spg347–\epg357.
  • [25] \auM. Shioya, A saturated stationary subset of $\mathcal{P}_{\kappa}\kappa^{+}$, \tiMath. Res. Lett., , 10 ((2003),)\spg493–\epg500.
  • [26] M. Shioya, Diamonds on $\mathcal{P}_{\kappa}\lambda$, Proceedings of Computational Prospects of Infinity, World Scientific, to appear.
  • [27] J. Steel, The Core Model Iterability Problem, Lecture Notes in Logic 8, Springer, Berlin, 1996.
  • [28] \auW. Woodin, Supercompact cardinals, sets of reals, and weakly homogeneous trees, \tiProc. Nat. Acad. Sci. U.S.A., , 85 ((1988),)\spg6587–\epg6591.
  • [29] W. Woodin, The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, de Gruyter Ser. in Logic and its Appl., 1, Walter de Gruyter, Berlin, 1999.