Advertisement

Israel Journal of Mathematics

, Volume 224, Issue 1, pp 343–366 | Cite as

Rigid ideals

  • Brent Cody
  • Monroe Eskew
Article
  • 21 Downloads

Abstract

An ideal I on a cardinal κ is called rigid if all automorphisms of P(κ)/I are trivial. An ideal is called μ-minimal if whenever GP(κ)/I is generic and XP(μ)V[G]V, it follows that V [X] = V [G]. We prove that the existence of a rigid saturated μ-minimal ideal on μ+, where μ is a regular cardinal, is consistent relative to the existence of large cardinals. The existence of such an ideal implies that GCH fails. However, we show that the existence of a rigid saturated ideal on μ+, where μ is an uncountable regular cardinal, is consistent with GCH relative to the existence of an almost-huge cardinal. Addressing the case μ = ω, we show that the existence of a rigid presaturated ideal on ω1 is consistent with CH relative to the existence of an almost-huge cardinal. The existence of a precipitous rigid ideal on μ+ where μ is an uncountable regular cardinal is equiconsistent with the existence of a measurable cardinal.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AS83]
    U. Abraham and S. Shelah, Forcing closed unbounded sets, Journal of Symbolic Logic 48 (1983), 643–657.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [BN15]
    O. Ben-Neria, The structure of the Mitchell order—II, Annals of Pure and Applied Logic 166 (2015), 1407–1432.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [BT82]
    J. E. Baumgartner and A. D. Taylor, Saturation properties of ideals in generic extensions. I, Transactions of the American Mathematical Society 270 (1982), 557–574.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [Cox]
    S. Cox, Layered posets and Kunen’s universal collapse, Notre Dame Journal of Formal Logic, to appear.Google Scholar
  5. [Cum10]
    J. Cummings, Iterated forcing and elementary embeddings, in Handbook of Set Theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 775–883.CrossRefGoogle Scholar
  6. [CZ14]
    S. Cox and M. Zeman, Ideal projections and forcing projections, Journal of Symbolic Logic 79 (2014), 1247–1285.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [FM09]
    S.-D. Friedman and M. Magidor, The number of normal measures, Journal of Symbolic Logic 74 (2009), 1069–1080.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [FMS88]
    M. Foreman, M. Magidor and S. Shelah, Martin’s maximum, saturated ideals, and nonregular ultrafilters. I, Annals of Mathematics 127 (1988), 1–47.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [For10]
    M. Foreman, Ideals and generic elementary embeddings, in Handbook of Set Theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 885–1147.CrossRefGoogle Scholar
  10. [For13]
    M. Foreman, Calculating quotient algebras of generic embeddings, Israel Journal of Mathematics 193 (2013), 309–341.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [Kun78]
    K. Kunen, Saturated ideals, Journal of Symbolic Logic 43 (1978), 65–76.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Kun83]
    K. Kunen, Set Theory, Studies in Logic and the Foundations of Mathematics, Vol. 102, North-Holland Publishing Co., Amsterdam, 1983.Google Scholar
  13. [Lar02]
    P. Larson, A uniqueness theorem for iterations, Journal of Symbolic Logic 67 (2002), 1344–1350.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [ST71]
    R. M. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin’s problem, Annals of Mathematics 94 (1971), 201–245.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [Woo10]
    W. H. Woodin, The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, De Gruyter Series in Logic and its Applications, Vol. 1, Walter de Gruyter GmbH & Co. KG, Berlin, 2010.Google Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsVirginia Commonwealth UniversityRichmondUSA

Personalised recommendations