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Stationary Sets

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Handbook of Set Theory

Abstract

Stationary sets play a fundamental role in modern set theory. This chapter attempts to explain this role and to describe the structure of stationary sets of ordinals and their generalization.

In the first part we develop the theory of closed unbounded and stationary subsets of a regular uncountable cardinal. The closed unbounded sets generate the closed unbounded filter. The dual ideal is the nonstationary ideal.

Among properties of stationary sets discussed in the article are reflection and saturation. Both are closely related to large cardinal properties.

In the second part we consider the generalization from stationary sets of ordinals to stationary sets of models. These play an essential role in proper forcing, discussed elsewhere in the Handbook.

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Bibliography

  1. Uri Abraham and Saharon Shelah. Forcing closed unbounded sets. The Journal of Symbolic Logic, 48(3):643–657, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  2. Pavel S. Aleksandrov and Pavel S. Urysohn. Mémoire sur les espaces topologiques compacts. Koninklijke Nederlandse Akademie van Wetenschappen te Amsterdam, 14:1–96, 1929.

    Google Scholar 

  3. Bohuslav Balcar, Thomas J. Jech, and Jindřich Zapletal. Semi-Cohen Boolean algebras. Annals of Pure and Applied Logic, 87(3):187–208, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  4. Stewart Baldwin. Generalizing the Mahlo hierarchy, with applications to the Mitchell models. Annals of Pure and Applied Logic, 25(2):103–127, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  5. Stewart Baldwin. The consistency strength of certain stationary subsets of  \(\mathcal{P}\sb{\kappa}\lambda\) . Proceedings of the American Mathematical Society, 92(1):90–92, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  6. James E. Baumgartner. A new class of order types. Annals of Mathematical Logic, 9(3):187–222, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  7. James E. Baumgartner. Applications of the proper forcing axiom. In Kenneth Kunen and Jerry E. Vaughan, editors, Handbook of Set-Theoretic Topology, pages 913–959. North-Holland, Amsterdam, 1984.

    Google Scholar 

  8. 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 

  9. James E. Baumgartner and Karel L. Prikry. On a theorem of Silver. Discrete Mathematics, 14(1):17–21, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  10. James E. Baumgartner and Karel L. Prikry. Singular cardinals and the Generalized Continuum Hypothesis. The American Mathematical Monthly, 84(2):108–113, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  11. James E. Baumgartner and Alan D. Taylor. Saturation properties of ideals in generic extensions. I. Transactions of the American Mathematical Society, 270(2):557–574, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  12. James E. Baumgartner, Leo A. Harrington, and Eugene M. Kleinberg. Adding a closed unbounded set. The Journal of Symbolic Logic, 41(2):481–482, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  13. James E. Baumgartner, Alan D. Taylor, and Stanley Wagon. On splitting stationary subsets of large cardinals. The Journal of Symbolic Logic, 42(2):203–214, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  14. Robert E. Beaudoin. Strong analogues of Martin’s axiom imply axiom R. The Journal of Symbolic Logic, 52(1):216–218, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  15. Mohamed Bekkali. Topics in Set Theory, volume 1476 of Lecture Notes in Mathematics. Springer, Berlin, 1991.

    MATH  Google Scholar 

  16. Gérard Bloch. Sur les ensembles stationnaires de nombres ordinaux et les suites distinguées de fonctions régressives. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, 236:265–268, 1953.

    MATH  MathSciNet  Google Scholar 

  17. Douglas R. Burke and Yo Matsubara. The extent of strength in the club filters. Israel Journal of Mathematics, 114:253–263, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  18. Donna M. Carr. The minimal normal filter on P κ λ. Proceedings of the American Mathematical Society, 86(2):316–320, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  19. Carlos A. Di Prisco and Wiktor Marek. Some properties of stationary sets. Dissertationes Mathematicae (Rozprawy Matematyczne), 218:37, 1983.

    MathSciNet  Google Scholar 

  20. Hans-Dieter Donder, Peter Koepke, and Jean-Pierre Levinski. Some stationary subsets of ℘(λ). Proceedings of the American Mathematical Society, 102(4):1000–1004, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  21. Qi Feng and Thomas J. Jech. Local clubs, reflection, and preserving stationary sets. Proceedings of the London Mathematical Society (3), 58(2):237–257, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  22. Qi Feng and Thomas J. Jech. Projective stationary sets and a strong reflection principle. Journal of the London Mathematical Society (2), 58(2):271–283, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  23. William G. Fleissner. Left separated spaces with point-countable bases. Transactions of the American Mathematical Society, 294(2):665–677, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  24. Géza Fodor. Eine Bemerkung zur Theorie der regressiven Funktionen. Acta Universitatis Szegediesis. Acta Scientiarum Mathematicarum, 17:139–142, 1956.

    MATH  MathSciNet  Google Scholar 

  25. Matthew Foreman and Menachem Magidor. Large cardinals and definable counterexamples to the Continuum Hypothesis. Annals of Pure and Applied Logic, 76(1):47–97, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  26. Matthew Foreman and Menachem Magidor. Mutually stationary sequences of sets and the non-saturation of the non-stationary ideal on P κ (λ). Acta Mathematica, 186(2):271–300, 2001.

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  28. Sakaé Fuchino. Some remarks on openly generated Boolean algebras. The Journal of Symbolic Logic, 59(1):302–310, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  29. Fred Galvin and András Hajnal. Inequalities for cardinal powers. Annals of Mathematics (2), 101:491–498, 1975.

    Article  MathSciNet  Google Scholar 

  30. Fred Galvin, Thomas Jech, and Menachem Magidor. An ideal game. The Journal of Symbolic Logic, 43(2):284–292, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  31. Moti Gitik. The nonstationary ideal on ℵ 2. Israel Journal of Mathematics, 48(4):257–288, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  32. Moti Gitik. Nonsplitting subset of P κ (κ +). The Journal of Symbolic Logic, 50(4):881–894, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  33. Moti Gitik. Changing cofinalities and the nonstationary ideal. Israel Journal of Mathematics, 56(3):280–314, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  34. Moti Gitik. On precipitousness of the nonstationary ideal over a supercompact. The Journal of Symbolic Logic, 51(3):648–662, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  35. Moti Gitik. Some results on the nonstationary ideal. Israel Journal of Mathematics, 92(1–3):61–112, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  36. Moti Gitik. Some results on the nonstationary ideal. II. Israel Journal of Mathematics, 99(1–3):175–188, 1997.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  38. Moti Gitik and Jiří Witzany. Consistency strength of the axiom of full reflection at large cardinals. Israel Journal of Mathematics, 93:113–124, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  39. Noa Goldring. The entire NS ideal on \(\cal P\sb\gamma\mu\) can be precipitous. The Journal of Symbolic Logic, 62(4):1161–1172, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  40. 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 

  41. Leo A. Harrington and Saharon Shelah. Some exact equiconsistency results in set theory. Notre Dame Journal of Formal Logic, 26(2):178–188, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  42. Lutz Heindorf and Leonid B. Shapiro. Nearly Projective Boolean Algebras, volume 1596 of Lecture Notes in Mathematics. Springer, Berlin, 1994. With an appendix by Sakaé Fuchino.

    MATH  Google Scholar 

  43. Thomas J. Jech. The closed unbounded filter over P κ (λ). Notices of the American Mathematical Society, 18:663, 1971. Abstract.

    Google Scholar 

  44. Thomas J. Jech. Some combinatorial problems concerning uncountable cardinals. Annals of Mathematical Logic, 5:165–198, 1972/73.

    Article  MathSciNet  Google Scholar 

  45. Thomas J. Jech. An infinite game played with stationary sets. Notices of the American Mathematical Society, 23:286, 1976. Abstract.

    Google Scholar 

  46. Thomas J. Jech. Precipitous ideals. In Logic Colloquium, vol. 76 (Oxford, 1976), volume 87 of Studies in Logic and the Foundations of Mathematics, pages 521–530. North-Holland, Amsterdam, 1977.

    Google Scholar 

  47. Thomas J. Jech. Stationary subsets of inaccessible cardinals. In James E. Baumgartner, Donald A. Martin, and Saharon Shelah, editors, Axiomatic Set Theory, volume 31 of Contemporary Mathematics, pages 115–142. American Mathematical Society, Providence, 1984.

    Google Scholar 

  48. Thomas J. Jech. Multiple Forcing, volume 88 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1986.

    MATH  Google Scholar 

  49. Thomas J. Jech. Set Theory. Springer Monographs in Mathematics. Springer, Berlin, 2002. The third millennium edition, revised and expanded.

    Google Scholar 

  50. Thomas J. Jech and Karel L. Prikry. On ideals of sets and the power set operation. Bulletin of the American Mathematical Society, 82(4):593–595, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  51. Thomas J. Jech and Karel L. Prikry. Ideals over uncountable sets: Applications of almost disjoint functions and generic ultrapowers. Memoirs of the American Mathematical Society, 18(214), 1979.

    Google Scholar 

  52. Thomas J. Jech and Saharon Shelah. A note on canonical functions. Israel Journal of Mathematics, 68(3):376–380, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  53. Thomas J. Jech and Saharon Shelah. Full reflection of stationary sets below ℵ ω . The Journal of Symbolic Logic, 55(2):822–830, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  54. Thomas J. Jech and Saharon Shelah. Full reflection of stationary sets at regular cardinals. American Journal of Mathematics, 115(2):435–453, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  55. Thomas J. Jech and Saharon Shelah. On reflection of stationary sets in ℘ κ λ. Transactions of the American Mathematical Society, 352(6):2507–2515, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  56. Thomas J. Jech and Jiří Witzany. Full reflection at a measurable cardinal. The Journal of Symbolic Logic, 59(2):615–630, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  57. Thomas J. Jech and W. Hugh Woodin. Saturation of the closed unbounded filter on the set of regular cardinals. Transactions of the American Mathematical Society, 292(1):345–356, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  58. Thomas J. Jech, Menachem Magidor, William J. Mitchell, and Karel L. Prikry. Precipitous ideals. The Journal of Symbolic Logic, 45(1):1–8, 1980.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  60. Chris A. Johnson. On saturated ideals and \(P\sb\kappa\lambda\) . Fundamenta Mathematicae, 129(3):215–221, 1988.

    MATH  MathSciNet  Google Scholar 

  61. Yuzuru Kakuda. Precipitousness of the ideal of thin sets on a measurable cardinal. In Logic Symposia, Hakone 1979, 1980 (Hakone, 1979/1980), volume 891 of Lecture Notes in Mathematics, pages 49–56. Springer, Berlin, 1981.

    Chapter  Google Scholar 

  62. Akihiro Kanamori. The Higher Infinite. Springer Monographs in Mathematics. Springer, Berlin, 2003. Second edition.

    MATH  Google Scholar 

  63. Jussi Ketonen. Some combinatorial principles. Transactions of the American Mathematical Society, 188:387–394, 1974.

    Article  MATH  MathSciNet  Google Scholar 

  64. Sabine Koppelberg. Projective Boolean algebras. In Handbook of Boolean Algebras, vol. 3, pages 741–773. North-Holland, Amsterdam, 1989.

    Google Scholar 

  65. Sabine Koppelberg. Characterizations of Cohen algebras. In Susan Andima et al., editors, Papers on General Topology and Applications, volume 704 of Annals of the New York Academy of Sciences, pages 222–237. New York Academy of Sciences, New York, 1993.

    Google Scholar 

  66. David W. Kueker. Löwenheim-Skolem and interpolation theorems in infinitary languages. Bulletin of the American Mathematical Society, 78:211–215, 1972.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  68. Kenneth Kunen. Saturated ideals. The Journal of Symbolic Logic, 43(1):65–76, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  69. Paul B. Larson. The Stationary Tower. Notes on a Course by W. Hugh Woodin, volume 32 of University Lecture Series. American Mathematical Society, Providence, 2004.

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  71. Paul Mahlo. Über lineare transfinite Mengen. Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematische-Physische Klass, 63:187–225, 1911.

    Google Scholar 

  72. Yo Matsubara. Menas’ conjecture and generic ultrapowers. Annals of Pure and Applied Logic, 36(3):225–234, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  73. Yo Matsubara. Splitting P κ λ into stationary subsets. The Journal of Symbolic Logic, 53(2):385–389, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  74. Yo Matsubara. Consistency of Menas’ conjecture. Journal of the Mathematical Society of Japan, 42(2):259–263, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  75. Yo Matsubara and Masahiro Shioya. Nowhere precipitousness of some ideals. The Journal of Symbolic Logic, 63(3):1003–1006, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  76. Alan H. Mekler and Saharon Shelah. The consistency strength of “Every stationary set reflects”. Israel Journal of Mathematics, 67(3):353–366, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  77. Telis K. Menas. On strong compactness and supercompactness. Annals of Mathematical Logic, 7:327–359, 1974/75.

    Article  MathSciNet  Google Scholar 

  78. Evgeny V. Ščepin. Functors and uncountable degrees of compacta. Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 36(3(219)):3–62, 255, 1981.

    Google Scholar 

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

    MATH  Google Scholar 

  80. Saharon Shelah. Around Classification Theory of Models, volume 1182 of Lecture Notes in Mathematics. Springer, Berlin, 1986.

    Book  MATH  Google Scholar 

  81. Saharon Shelah. Iterated forcing and normal ideals on \(\omega\sb1\) . Israel Journal of Mathematics, 60(3):345–380, 1987.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  83. Masahiro Shioya. A saturated stationary subset of \(\mathcal{P}\sb\kappa \kappa^{+}\) . Mathematical Research Letters, 10(4):493–500, 2003.

    MATH  MathSciNet  Google Scholar 

  84. Jack H. Silver. On the singular cardinals problem. In Proceedings of the International Congress of Mathematicians, pages 265–268. Canadian Mathematical Congress, Montreal, 1975.

    Google Scholar 

  85. Robert M. Solovay. Real-valued measurable cardinals. In Dana S. Scott, editor, Axiomatic Set Theory, pages 397–428. American Mathematical Society, Providence, 1971.

    Google Scholar 

  86. John R. Steel. The Core Model Iterability Problem, volume 8 of Lecture Notes in Logic. Springer, Berlin, 1996.

    MATH  Google Scholar 

  87. John R. Steel and Robert van Wesep. Two consequences of determinacy consistent with choice. Transactions of the American Mathematical Society, 272(1):67–85, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  88. Stevo Todorčević. On a conjecture of R. Rado. Journal of the London Mathematical Society (2), 27(1):1–8, 1983.

    Article  MATH  Google Scholar 

  89. Stevo Todorčević. Reflecting stationary sets, 1985. Handwritten notes.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  91. Jiří Witzany. Possible behaviours of the reflection ordering of stationary sets. The Journal of Symbolic Logic, 60(2):534–547, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  92. W. Hugh Woodin. Some consistency results in ZFC using AD. In Cabal seminar 79–81, volume 1019 of Lecture Notes in Mathematics, pages 172–198. Springer, Berlin, 1983.

    Chapter  Google Scholar 

  93. W. Hugh Woodin. Supecompact cardinals, sets of reals, and weakly homogeneous trees. Proceedings of the National Academy USA, 85(18):6587–6591, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  94. W. Hugh Woodin. The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, volume 1 of de Gruyter Series in Logic and Its Applications. de Gruyter, Berlin, 1999.

    MATH  Google Scholar 

  95. Yasuo Yoshinobu. On strength of precipitousness of some ideals and towers. Journal of the Mathematical Society of Japan, 51(3):535–541, 1999.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Center for Theoretical Study, Prague, 1, Czech Republic

    Thomas Jech

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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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Jech, T. (2010). Stationary Sets. In: Foreman, M., Kanamori, A. (eds) Handbook of Set Theory. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5764-9_2

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