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Successors of Singular Cardinals

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Handbook of Set Theory

Abstract

Successors of singular cardinals are a peculiar—although they are successor cardinals, they can still exhibit some of the behaviors typically associated with large cardinals. In this chapter, we examine the combinatorics of successors of singular cardinals in detail. We use stationary reflection as our point of entry into the subject, and we sketch Magidor’s proof that it is consistent that all stationary subsets of such a cardinal reflect. Further consideration of Magidor’s proof brings us to Shelah’s ideal I[λ] and the related Approachability Property (AP); we give a fairly comprehensive treatment of these topics. Building on this, we then turn to squares, scales, and the influence these objects exert on questions of pertaining to reflection phenomena. The chapter concludes with a brief look at square-brackets partition relations and their relation to club-guessing principles.

The author acknowledges support from NSF grant DMS-0506063 during the preparation of this chapter.

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Bibliography

  1. Uri Abraham and Menachem Magidor. Cardinal arithmetic. Chapter 14 in this Handbook. 10.1007/978-1-4020-5764-9_15.

  2. Arthur W. Apter and James Cummings. A global version of a theorem of Ben-David and Magidor. Annals of Pure and Applied Logic, 102(3):199–222, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  3. Arthur W. Apter and James M. Henle. On box, weak box and strong compactness. The Bulletin of the London Mathematical Society, 24(6):513–518, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  4. Zoltan T. Balogh, Sheldon W. Davis, Winfried Just, Saharon Shelah, and Paul J. Szeptycki. Strongly almost disjoint sets and weakly uniform bases. Transactions of the American Mathematical Society, 352(11):4971–4987, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  5. James E. Baumgartner. A new class of order types. Annals of Pure and Applied Logic, 9(3):187–222, 1976.

    MATH  MathSciNet  Google Scholar 

  6. James E. Baumgartner. On the size of closed unbounded sets. Annals of Pure and Applied Logic, 54(3):195–227, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  7. Shai Ben-David and Menachem Magidor. The weak □ * is really weaker than the full □. The Journal of Symbolic Logic, 51:1029–1033, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  8. Shai Ben-David and Saharon Shelah. The two-cardinals transfer property and resurrection of supercompactness. Proceedings of the American Mathematical Society, 124(9):2827–2837, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  9. Lev Bukovský and Eva Copláková-Hartová. Minimal collapsing extensions of models of ZFC. Annals of Pure and Applied Logic, 46(3):265–298, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  10. Maxim R. Burke and Menachem Magidor. Shelah’s pcf theory and its applications. Annals of Pure and Applied Logic, 50(3):207–254, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen-Chung Chang. Descendingly incomplete ultrafilters. Transactions of the American Mathematical Society, 126:108–118, 1967.

    Article  MATH  MathSciNet  Google Scholar 

  12. James Cummings. Collapsing successors of singulars. Proceedings of the American Mathematical Society, 125(9):2703–2709, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  13. James Cummings. Large cardinal properties of small cardinals. In Set Theory (Curaçao, 1995; Barcelona, 1996), pages 23–39. Kluwer Academic, Dordrecht, 1998.

    Google Scholar 

  14. James Cummings. Notes on singular cardinal combinatorics. Notre Dame Journal of Formal Logic, 46(3):251–282, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  15. James Cummings and Ernest Schimmerling. Indexed squares. Israel Journal of Mathematics, 131:61–99, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  16. James Cummings, Matthew Foreman, and Menachem Magidor. Squares, scales and stationary reflection. Journal of Mathematical Logic, 1(1):35–98, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  17. James Cummings, Matthew Foreman, and Menachem Magidor. Canonical structure in the universe of set theory I. Annals of Pure and Applied Logic, 129(1–3):211–243, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  18. James Cummings, Matthew Foreman, and Menachem Magidor. Canonical structure in the universe of set theory II. Annals of Pure and Applied Logic, 142(1–3):55–75, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  19. Keith J. Devlin. Constructibility. Perspectives in Mathematical Logic. Springer, Berlin, 1984.

    MATH  Google Scholar 

  20. Keith J. Devlin and Håvard Johnsbråten. The Souslin Problem, volume 405 of Lecture Notes in Mathematics. Springer, Berlin, 1974.

    MATH  Google Scholar 

  21. Mirna Džamonja and Saharon Shelah. On squares, outside guessing of clubs and \(I\sb {<f}[\lambda]\) . Fundamenta Mathematicae, 148(2):165–198, 1995.

    MATH  MathSciNet  Google Scholar 

  22. Todd Eisworth. A note on Jónsson cardinals. In Proceedings of the Spring Topology and Dynamical Systems Conference, Vol. 27, pages 173–178. 2003.

    Google Scholar 

  23. Todd Eisworth. On ideals related to I[λ]. Notre Dame Journal of Formal Logic, 46(3):301–307, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  24. Todd Eisworth. A note on negative square brackets partition relations. Fundamenta Mathematicae, accepted.

    Google Scholar 

  25. Todd Eisworth and Saharon Shelah. Successors of singular cardinals and coloring theorem I. Archive for Mathematical Logic, 44(5):597–618, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  26. Paul Erdős and András Hajnal. On a problem of B. Jónsson. Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, 14:19–23, 1966.

    Google Scholar 

  27. Paul Erdős, András Hajnal, and Richard Rado. Partition relations for cardinal numbers. Acta Mathematica Academiae Scientiarum Hungaricae, 16:93–196, 1965.

    Article  MathSciNet  Google Scholar 

  28. Paul Erdős, András Hajnal, Attila Máté, and Richard Rado. Combinatorial Set Theory: Partition Relations for Cardinals, volume 106 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1984.

    Google Scholar 

  29. William G. Fleissner. If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal. Transactions of the American Mathematical Society, 273(1):365–373, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  30. William G. Fleissner. Normal nonmetrizable Moore space from continuum hypothesis or nonexistence of inner models with measurable cardinals. Proceedings of the National Academy of Sciences USA, 79(4):1371–1372, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  31. Matthew Foreman. Potent axioms. Transactions of the American Mathematical Society, 294(1):1–28, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  32. Matthew Foreman. Some problems in singular cardinals combinatorics. Notre Dame Journal of Formal Logic, 46(3):309–322, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  33. Matthew Foreman and Peter Komjáth. The club guessing ideal: commentary on a theorem of Gitik and Shelah. Journal of Mathematical Logic, 5(1):99–147, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  34. Matthew Foreman and Menachem Magidor. A very weak square principle. The Journal of Symbolic Logic, 62(1):175–196, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  35. Matthew Foreman and Stevo Todorčević. A new Löwenheim-Skolem theorem. Transactions of the American Mathematical Society, 357(5):1693–1715, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  36. Matthew Foreman, Menachem Magidor, and Saharon Shelah. Martin’s maximum, saturated ideals, and nonregular ultrafilters. I. Annals of Mathematics (2), 127(1):1–47, 1988.

    Article  MathSciNet  Google Scholar 

  37. Fred Galvin and Karel L. Prikry. Infinitary Jonsson algebras and partition relations. Algebra Universalis, 6(3):367–376, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  38. Mofi Gitik. Prikry-type forcings. Chapter 16 in this Handbook. 10.1007/978-1-4020-5764-9_17.

  39. Moti Gitik and Assaf Sharon. On SCH and the approachability property. Proceedings of the American Mathematical Society, 136:311–320, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  40. Moti Gitik and Saharon Shelah. Less saturated ideals. Proceedings of the American Mathematical Society, 125(5):1523–1530, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  41. John Gregory. Higher Souslin trees and the generalized continuum hypothesis. The Journal of Symbolic Logic, 41(3):663–671, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  42. András Hajnal and Peter Hamburger. Set Theory, volume 48 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1999. Translated from the 1983 Hungarian original by Attila Máté.

    MATH  Google Scholar 

  43. András Hajnal, Istvan Juhász, and Saharon Shelah. Splitting strongly almost disjoint families. Transactions of the American Mathematical Society, 295(1):369–387, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  44. András Hajnal, Istvan Juhász, and Saharon Shelah. Strongly almost disjoint families, revisited. Fundamenta Mathematicae, 163(1):13–23, 2000.

    MATH  MathSciNet  Google Scholar 

  45. William P. Hanf and Dana S. Scott. Classifying inaccessible cardinals (abstract). Notices of the American Mathematical Society, 8:445, 1961.

    Google Scholar 

  46. Michael Holz, Karsten Steffens, and Edmund Weitz. Introduction to Cardinal Arithmetic. Birkhäuser Advanced Texts. Birkhäuser, Basel, 1999.

    MATH  Google Scholar 

  47. Tetsuya Ishiu. Club guessing sequences and filters. The Journal of Symbolic Logic, 70(4):1037–1071, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  48. Tetsuya Ishiu. A tail club guessing ideal can be saturated without being a restriction of the nonstationary ideal. Notre Dame Journal of Formal Logic, 46(3):327–333, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  49. Thomas J. Jech. Stationary Sets. Chapter 1 in this Handbook. 10.1007/978-1-4020-5764-9_2.

  50. Thomas J. Jech. Some combinatorial problems concerning uncountable cardinals. Annals of Pure and Applied Logic, 5:165–198, 1972/1973.

    MathSciNet  Google Scholar 

  51. Thomas J. Jech. Set Theory. Springer Monographs in Mathematics. Springer, Berlin, 2002. Third millennium edition.

    Google Scholar 

  52. Ronald B. Jensen. The fine structure of the constructible hierarchy. Annals of Mathematical Logic, 4(3):229–308, 1972.

    Article  MATH  Google Scholar 

  53. Akihiro Kanamori. The Higher Infinite. Perspectives in Mathematical Logic. Springer, Berlin, 2003. Second edition.

    MATH  Google Scholar 

  54. H. Jerome Keisler and Frederick Rowbottom. Constructible sets and weakly compact cardinals (abstract). Notices of the American Mathematical Society, 12:373–374, 1965.

    Google Scholar 

  55. Menachem Kojman. The ABC of PCF. http://www.cs.bgu.ac.il/kojman/ABCI.pdf.

  56. Menachem Kojman. Exact upper bounds and their uses in set theory. Annals of Pure and Applied Logic, 92(3):267–282, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  57. David W. Kueker. Countable approximations and Löwenheim-Skolem theorems. Annals of Pure and Applied Logic, 11(1):57–103, 1977.

    MATH  MathSciNet  Google Scholar 

  58. Kenneth Kunen. Elementary embeddings and infinitary combinatorics. The Journal of Symbolic Logic, 36:407–413, 1971.

    Article  MATH  MathSciNet  Google Scholar 

  59. Kenneth Kunen. Set Theory, an Introduction to Independence Proofs, volume 102 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1980.

    MATH  Google Scholar 

  60. Kenneth Kunen and Karel L. Prikry. On descendingly incomplete ultrafilters. The Journal of Symbolic Logic, 36:650–652, 1971.

    Article  MathSciNet  Google Scholar 

  61. Jean-Pierre Levinski, Menachem Magidor, and Saharon Shelah. Chang’s conjecture for \(\aleph\sb \omega\) . Israel Journal of Mathematics, 69(2):161–172, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  62. Menachem Magidor. Reflecting stationary sets. The Journal of Symbolic Logic, 47(4):755–771, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  63. Menachem Magidor. Aronszajn trees and weak squares. Unpublished notes 1988.

    Google Scholar 

  64. Menachem Magidor. Lectures on weak square principles and forcing axioms. Jerusalem Logic Seminar, 1995.

    Google Scholar 

  65. Menachem Magidor and Saharon Shelah. When does almost free imply free? Journal of the American Mathematical Society, 7(4):769–830, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  66. Menachem Magidor and Saharon Shelah. The tree property at successors of singular cardinals. Archive for Mathematical Logic, 35(5–6):385–404, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  67. Eric C. Milner and Saharon Shelah. Some theorems on transversals. In Infinite and Finite Sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. III, volume 10 of Colloquia Mathematica Societatis János Bolyai, pages 1115–1126. North-Holland, Amsterdam, 1975.

    Google Scholar 

  68. William J. Mitchell. Adding closed unbounded subsets of \(\omega\sb 2\) with finite forcing. Notre Dame Journal of Formal Logic, 46(3):357–371, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  69. William J. Mitchell. I[ω 2] can be the nonstationary ideal on Cof(ω 1). Preprint, June 2005.

    Google Scholar 

  70. Peter J. Nyikos. A provisional solution to the normal Moore space problem. Proceedings of the American Mathematical Society, 78(3):429–435, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  71. Karel L. Prikry. On descendingly complete ultrafilters. In Cambridge Summer School in Mathematical Logic (1971), volume 337 of Lecture notes in Mathematics. pages 459–488. Springer, Berlin, 1973.

    Chapter  Google Scholar 

  72. Frederick Rowbottom. Some strong axioms of infinity incompatible with the axiom of constructibility. Annals of Pure and Applied Logic, 3(1):1–44, 1971.

    MATH  MathSciNet  Google Scholar 

  73. Ernest Schimmerling. Combinatorial principles in the core model for one Woodin cardinal. Annals of Pure and Applied Logic, 74(2):153–201, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  74. Ernest Schimmerling. Combinatorial set theory and inner models. In Set Theory (Curaçao 1995; Barcelona 1996), pages 207–212. Kluwer Academic, Dordrecht, 1998.

    Google Scholar 

  75. Ernest Schimmerling. A finite family weak square principle. The Journal of Symbolic Logic, 64(3):1087–1110, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  76. Ernest Schimmerling. A question about Suslin trees and the weak square hierarchy. Notre Dame Journal of Formal Logic, 46(3):373–374, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  77. Ernest Schimmerling and Martin Zeman. Square in core models. The Bulletin of Symbolic Logic, 7(3):305–314, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  78. Ernest Schimmerling and Martin Zeman. Characterization of \(\square\sb \kappa\) in core models. Journal of Mathematical Logic, 4(1):1–72, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  79. Assaf Sharon. Weak squares, scales, stationary reflection, and the failure of SCH. PhD thesis, Tel Aviv University, 2005.

    Google Scholar 

  80. Saharon Shelah. Diamonds. No. 922 on Shelah’s list.

    Google Scholar 

  81. Saharon Shelah. A compactness theorem for singular cardinals, free algebras, Whitehead problem, and transversals. Israel Journal of Mathematics, 21:319–339, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  82. Saharon Shelah. Jonsson algebras in successor cardinals. Israel Journal of Mathematics, 30(1–2):57–64, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  83. Saharon Shelah. On the successors of singular cardinals. In Maurice Boffa, Dirk Van Dalen, and Kenneth McAloon, editors, Logic Colloquium 78, volume 97 of Studies in Logic and the Foundations of Mathematics, pages 357–380. North-Holland, Amsterdam, 1978.

    Chapter  Google Scholar 

  84. Saharon Shelah. Proper Forcing, volume 940 of Lecture Notes in Mathematics. Springer, Berlin, 1982.

    MATH  Google Scholar 

  85. Saharon Shelah. Diamonds, uniformization. The Journal of Symbolic Logic, 49(4):1022–1033, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  86. Saharon Shelah. Was Sierpiński right? I. Israel Journal of Mathematics, 62(3):355–380, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  87. Saharon Shelah. Reflecting stationary sets and successors of singular cardinals. Archive for Mathematical Logic, 31(1):25–53, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  88. Saharon Shelah. Advances in cardinal arithmetic. In Finite and Infinite Combinatorics in Sets and Logic (Banff, AB, 1991), volume 411 of NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, pages 355–383. Kluwer Academic, Dordrecht, 1993.

    Google Scholar 

  89. Saharon Shelah. Cardinal Arithmetic, volume 29 of Oxford Logic Guides. Clarendon, Oxford, 1994.

    MATH  Google Scholar 

  90. Saharon Shelah. Lectures on I[λ]. Jerusalem Logic Seminar, 1998.

    Google Scholar 

  91. Saharon Shelah. On what I do not understand (and have something to say). I. Fundamenta Mathematicae, 166(1–2):1–82, 2000.

    MATH  MathSciNet  Google Scholar 

  92. Saharon Shelah. More Jonsson algebras. Archive for Mathematical Logic, 42(1):1–44, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  93. Saharon Shelah. Successor of singulars: combinatorics and not collapsing cardinals ≤κ in (<κ)-support iterations. Israel Journal of Mathematics, 134:127–155, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  94. Robert M. Solovay. Strongly compact cardinals and the GCH. In Leon Henken et al., editors, Proceedings of the Tarski Symposium, volume XXV of Proceedings of Symposia in Pure Mathematics, pages 365–372. American Mathematical Society, Providence, 1974.

    Google Scholar 

  95. Robert M. Solovay, William N. Reinhardt, and Akihiro Kanamori. Strong axioms of infinity and elementary embeddings. Annals of Mathematical Logic, 13(1):73–116, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  96. John R. Steel. PFA implies ADL(ℝ). The Journal of Symbolic Logic, 70(4):1255–1296, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  97. Stevo Todorčević. Partitioning pairs of countable ordinals. Acta Mathematica, 159(3–4):261–294, 1987.

    MATH  MathSciNet  Google Scholar 

  98. Stevo Todorčević. Coherent sequences. Chapter 3 in this Handbook. 10.1007/978-1-4020-5764-9_4.

  99. Jan Tryba. On Jónsson cardinals with uncountable cofinality. Israel Journal of Mathematics, 49(4):315–324, 1984.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics, Ohio University, Athens, OH, 45701, USA

    Todd Eisworth

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  1. Todd Eisworth
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  1. School of Physical Sciences, University of California, Irvine, Irvine, 92697-3875, U.S.A.

    Matthew Foreman

  2. Dept. Manufacturing Engineering, Boston University, St. Mary's St. 15, Boston, 02215, U.S.A.

    Akihiro Kanamori

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Eisworth, T. (2010). Successors of Singular Cardinals. In: Foreman, M., Kanamori, A. (eds) Handbook of Set Theory. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5764-9_16

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