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Handbook of Set Theory pp 1497–1594Cite as

The Covering Lemma

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Abstract

Our understanding of inner models was transformed in 1974 by as set of handwritten notes of Ronald Jensen with the modest title “Marginalia on a Theorem of Silver”. Before the covering lemma, Gödel’s class L of constructible sets was a model of set theory of which much was know, but which had little connection with the universe; with the covering lemma it is, in the absence of large cardinals, the supporting skeleton of the universe.

The first section of this chapter discusses the forms of the covering lemma, beginning with Jensen’s original lemma for L, going to extensions taking into account arbitrary sequences of measurable cardinals, and then describing variants of the statement: the weak covering lemma, which is basic to inner models beyond measurable cardinals, the strong covering lemma, and variants such as that of Magidor which avoid the need for second order closure. Later sections discuss applications, give proofs of the statements, and discuss what is known up to and beyond a Woodin cardinal.

The writing of this chapter was partially supported by grant number DMS-9970536 from the National Science Foundation.

The author would also like to thank Diego Rojas-Robello, Assaf Sharon and the referee for helpful suggestions.

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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. Stewart Baldwin. Between strong and superstrong. The Journal of Symbolic Logic, 51(3):547–559, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  3. James E. Baumgartner. Ineffability properties of cardinals. II. In Logic, Foundations of Mathematics and Computability Theory (Proc. Fifth Internat. Congr. Logic Methodology Philos. Sci., Univ. Western Ontario, London, ON, 1975), Part I, pages 87–1069. Reidel, Dordrecht, 1977.

    Google Scholar 

  4. Paul J. Cohen. The independence of the continuum hypothesis, I. Proceedings of the National Academy of Sciences USA, 50:1143–1148, 1963.

    Article  Google Scholar 

  5. Paul J. Cohen. The independence of the continuum hypothesis, II. Proceedings of the National Academy of Sciences USA, 50:105–110, 1964.

    Article  Google Scholar 

  6. Keith J. Devlin. Aspects of Constructibility, volume 354 of Lecture Notes in Mathematics. Springer, Berlin, 1973.

    MATH  Google Scholar 

  7. Keith J. Devlin and Ronald B. Jensen. Marginalia to a theorem of Silver. In Proceedings of the ISILC Logic Conference (Kiel 1974), volume 499 of Lecture Notes in Mathematics, pages 115–142. Springer, Berlin, 1975.

    Google Scholar 

  8. Anthony J. Dodd. The Core Model, volume 61 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1982.

    MATH  Google Scholar 

  9. Anthony J. Dodd and Ronald B. Jensen. The core model. Annals of Mathematical Logic, 20(1):43–75, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  10. Anthony J. Dodd and Ronald B. Jensen. The covering lemma for K. Annals of Mathematical Logic, 22(1):1–30, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  11. Anthony J. Dodd and Ronald B. Jensen. The covering lemma for L[U]. Annals of Mathematical Logic, 22(2):127–135, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  12. Hans-Dieter Donder, Ronald B. Jensen, and Bernd J. Koppelberg. Some applications of the core model. In Set Theory and Model Theory (Bonn, 1979), volume 872 of Lecture Notes in Mathematics, pages 55–97. Springer, Berlin, 1981.

    Chapter  Google Scholar 

  13. William B. Easton. Powers of regular cardinals. PhD thesis, Princeton University, 1964.

    Google Scholar 

  14. William B. Easton. Powers of regular cardinals. Annals of Mathematical Logic, 1:139–178, 1970.

    Article  MATH  MathSciNet  Google Scholar 

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

  16. Moti Gitik. The negation of the singular cardinal hypothesis from o(κ)=κ ++. Annals of Pure and Applied Logic, 43(3):209–234, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  17. Moti Gitik. The strength of the failure of the singular cardinal hypothesis. Annals of Pure and Applied Logic, 51(3):215–240, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  18. Moti Gitik. On measurable cardinals violating the continuum hypothesis. Annals of Pure and Applied Logic, 63(3):227–240, 1993.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Moti Gitik. Blowing up the power of a singular cardinal. Annals of Pure and Applied Logic, 80(1):17–33, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  21. Moti Gitik. On hidden extenders. Archive for Mathematical Logic, 35(5–6):349–369, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  22. Moti Gitik. Some results on the nonstationary ideal. II. Israel Journal of Mathematics, 99:175–188, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  23. Moti Gitik and William J. Mitchell. Indiscernible sequences for extenders, and the singular cardinal hypothesis. Annals of Pure and Applied Logic, 82(3):273–316, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  24. Moti Gitik, Ralf Schindler, and Saharon Shelah. Pcf theory and Woodin cardinals. In Logic Colloquium’02, volume 27 of Lecture Notes in Logic, pages 172–205. Association for Symbolic Logic, Urbana, 2006.

    Google Scholar 

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

    Google Scholar 

  26. Ronald B. Jensen. Marginalia to a theorem of Silver. Handwritten notes (several sets), 1974

    Google Scholar 

  27. Ronald B. Jensen, Ernest Schimmerling, Ralf Schindler, and John R. Steel. Stacking mice. The Journal of Symbolic Logic, 74(1):315–335, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  28. Akihiro Kanamori. The Higher Infinite: Large Cardinals in Set Theory. Springer, Berlin, 2003. Second edition.

    MATH  Google Scholar 

  29. Benedikt Löwe and John R. Steel. An introduction to core model theory. In Sets and Proofs (Leeds, 1997), volume 258 of London Mathematical Society Lecture Note Series, pages 103–157. Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  30. Menachem Magidor. Changing cofinality of cardinals. Fundamenta Mathematicae, 99(1):61–71, 1978.

    MATH  MathSciNet  Google Scholar 

  31. Menachem Magidor. Representing sets of ordinals as countable unions of sets in the core model. Transactions of the American Mathematical Society, 317(1):91–126, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  32. Wilmiam J. Mitchell. Beginning inner model theory. Chapter 17 in this Handbook. 10.1007/978-1-4020-5764-9_18.

  33. William J. Mitchell. Ramsey cardinals and constructibility. Journal of Symbolic Logic, 44(2):260–266, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  34. William J. Mitchell. How weak is a closed unbounded ultrafilter? In Dirk van Dalen, Daniel Lascar, and Timothy J. Smiley, editors, Logic Colloquium ’80 (Prague, 1998), pages 209–230. North-Holland, Amsterdam, 1982.

    Google Scholar 

  35. William J. Mitchell. The core model for sequences of measures. I. Mathematical Proceedings of the Cambridge Philosophical Society, 95(2):229–260, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  36. William J. Mitchell. Indiscernibles, skies, and ideals. In James E. Baumgartner, Donald A. Martin, and Saharon Shelah, editors, Axiomatic Set Theory (Boulder, CO, 1983), volume 31 of Contemporary Mathematics, pages 161–182. American Mathematical Society, Providence, 1984.

    Google Scholar 

  37. William J. Mitchell. Applications of the covering lemma for sequences of measures. Transactions of the American Mathematical Society, 299(1):41–58, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  38. William J. Mitchell. Definable singularity. Transactions of the American Mathematical Society, 327(1):407–426, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  39. William J. Mitchell. \(\mathbf{\Sigma}^{1}_{3}\) -absoluteness for sequences of measures. In Set Theory of the Continuum (Berkeley, CA, 1989), volume 26 of Mathematical Sciences Research Institute Publications, pages 311–355. Springer, New York, 1992.

    Google Scholar 

  40. William J. Mitchell. On the singular cardinal hypothesis. Transactions of the American Mathematical Society, 329(2):507–530, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  41. William J. Mitchell. Jónsson cardinals, Erdős cardinals, and the core model. Preliminary, available on Logic Eprints, 1994.

    Google Scholar 

  42. William J. Mitchell. A hollow shell: Covering lemmas without a core. In Carlos Di Prisco, Jean A. Larson, Joan Bagaria, and Adrian R. D. Mathias, editors, Set Theory: Techniques and Applications, pages 183–198. Kluwer Academic, Dordrecht, 1998.

    Google Scholar 

  43. William J. Mitchell and Ernest Schimmerling. Weak covering without countable closure. Mathematical Research Letters, 2(5):595–609, 1995.

    MATH  MathSciNet  Google Scholar 

  44. William J. Mitchell and Ralf Schindler. A universal extender model without large cardinals in V. The Journal of Symbolic Logic, 69(2):371–386, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  45. William J. Mitchell, Ernest Schimmerling, and John R. Steel. The covering lemma up to a Woodin cardinal. Annals of Pure and Applied Logic, 84(2):219–255, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  46. Karl Prikry. Changing measurable into accessible cardinals. Dissertationes Mathematicae (Rozprawy Mathematycne), 68:359–378, 1971.

    MathSciNet  Google Scholar 

  47. Ernest Schimmerling. A core model toolbox and guide. Chapter 20 in this Handbook. 10.1007/978-1-4020-5764-9_21.

  48. Ernest Schimmerling and W. Hugh Woodin. The Jensen covering property. The Journal of Symbolic Logic, 66(4):1505–1523, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  49. Ralf Schindler. Weak covering and the tree property. Archive for Mathematical Logic, 38(8):515–520, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  50. Ralf Schindler. The core model for almost linear iterations. Annals of Pure and Applied Logic, 116(1–3):205–272, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  51. Ralf Schindler. Iterates of the core model. The Journal of Symbolic Logic, 71:241–251, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  52. Ralf Schindler and Martin Zeman. Fine structure. Chapter 9 in this Handbook. 10.1007/978-1-4020-5764-9_10.

  53. Saharon Shelah. Cardinal Arithmetic, volume 29 of Oxford Logic Guides. Oxford University Press, Oxford, 1994.

    MATH  Google Scholar 

  54. Saharon Shelah. Strong covering without squares. Fundamenta Mathematicae, 166(1–2):87–107, 2000. Saharon Shelah’s anniversary issue.

    MATH  MathSciNet  Google Scholar 

  55. Saharon Shelah and W. Hugh Woodin. Forcing the failure of CH by adding a real. The Journal of Symbolic Logic, 49(4):1185–1189, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  56. Jack Silver. On the singular cardinals problem. In Proceedings of the International Congress of Mathematicians, Vol. 1, pages 265–268. Canadian Mathematical Congress, Vancouver, 1975.

    Google Scholar 

  57. John R. Steel. An outline of inner model theory. Chapter 19 in this Handbook. 10.1007/978-1-4020-5764-9_20.

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

    MATH  Google Scholar 

  59. John R. Steel and Philip D. Welch. \(\Sigma^{1}_{3}\) absoluteness and the second uniform indiscernible. Israel Journal of Mathematics, 104:157–190, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  60. Claude Sureson. Excursion en mesurabilite. PhD thesis, City University of New York, 1984.

    Google Scholar 

  61. John Vickers and Philip D. Welch. On elementary embeddings from an inner model to the universe. The Journal of Symbolic Logic, 66(3):1090–1116, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  62. Philip D. Welch. Some remarks on the maximality of inner models. In Samuel R. Buss, Petr Hajek, and Pavel Pudlak, editors, Logic Colloquium ’98 (Prague), volume 13 of Lecture Notes in Logic, pages 516–540. Association Symbolic Logic, Urbana, 2000.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  64. W. Hugh Woodin. The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. de Gruyter, Berlin, 1999.

    MATH  Google Scholar 

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  1. Department of Mathematics, University of Florida, Gainsville, FL, 32611, USA

    William J. Mitchell

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  1. William J. Mitchell
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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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Mitchell, W.J. (2010). The Covering Lemma. In: Foreman, M., Kanamori, A. (eds) Handbook of Set Theory. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5764-9_19

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