The Covering Lemma



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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Uri Abraham and Menachem Magidor. Cardinal arithmetic. Chapter 14 in this Handbook. 10.1007/978-1-4020-5764-9_15.
  2. [2]
    Stewart Baldwin. Between strong and superstrong. The Journal of Symbolic Logic, 51(3):547–559, 1986. MATHCrossRefMathSciNetGoogle Scholar
  3. [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. [4]
    Paul J. Cohen. The independence of the continuum hypothesis, I. Proceedings of the National Academy of Sciences USA, 50:1143–1148, 1963. CrossRefGoogle Scholar
  5. [5]
    Paul J. Cohen. The independence of the continuum hypothesis, II. Proceedings of the National Academy of Sciences USA, 50:105–110, 1964. CrossRefGoogle Scholar
  6. [6]
    Keith J. Devlin. Aspects of Constructibility, volume 354 of Lecture Notes in Mathematics. Springer, Berlin, 1973. MATHGoogle Scholar
  7. [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. [8]
    Anthony J. Dodd. The Core Model, volume 61 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1982. MATHGoogle Scholar
  9. [9]
    Anthony J. Dodd and Ronald B. Jensen. The core model. Annals of Mathematical Logic, 20(1):43–75, 1981. MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Anthony J. Dodd and Ronald B. Jensen. The covering lemma for K. Annals of Mathematical Logic, 22(1):1–30, 1982. MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Anthony J. Dodd and Ronald B. Jensen. The covering lemma for L[U]. Annals of Mathematical Logic, 22(2):127–135, 1982. MATHCrossRefMathSciNetGoogle Scholar
  12. [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. CrossRefGoogle Scholar
  13. [13]
    William B. Easton. Powers of regular cardinals. PhD thesis, Princeton University, 1964. Google Scholar
  14. [14]
    William B. Easton. Powers of regular cardinals. Annals of Mathematical Logic, 1:139–178, 1970. MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Moti Gitik. Prikry-type forcings. Chapter 16 in this Handbook. 10.1007/978-1-4020-5764-9_17.
  16. [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. MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Moti Gitik. The strength of the failure of the singular cardinal hypothesis. Annals of Pure and Applied Logic, 51(3):215–240, 1991. MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Moti Gitik. On measurable cardinals violating the continuum hypothesis. Annals of Pure and Applied Logic, 63(3):227–240, 1993. MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Moti Gitik. Some results on the nonstationary ideal. Israel Journal of Mathematics, 92(1–3):61–112, 1995. MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    Moti Gitik. Blowing up the power of a singular cardinal. Annals of Pure and Applied Logic, 80(1):17–33, 1996. MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Moti Gitik. On hidden extenders. Archive for Mathematical Logic, 35(5–6):349–369, 1996. MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    Moti Gitik. Some results on the nonstationary ideal. II. Israel Journal of Mathematics, 99:175–188, 1997. MATHCrossRefMathSciNetGoogle Scholar
  23. [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. MATHCrossRefMathSciNetGoogle Scholar
  24. [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. [25]
    Thomas J. Jech. Set Theory. Springer Monographs in Mathematics. Springer, Berlin, 2002. The third millennium edition, revised and expanded. Google Scholar
  26. [26]
    Ronald B. Jensen. Marginalia to a theorem of Silver. Handwritten notes (several sets), 1974 Google Scholar
  27. [27]
    Ronald B. Jensen, Ernest Schimmerling, Ralf Schindler, and John R. Steel. Stacking mice. The Journal of Symbolic Logic, 74(1):315–335, 2009. MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    Akihiro Kanamori. The Higher Infinite: Large Cardinals in Set Theory. Springer, Berlin, 2003. Second edition. MATHGoogle Scholar
  29. [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. [30]
    Menachem Magidor. Changing cofinality of cardinals. Fundamenta Mathematicae, 99(1):61–71, 1978. MATHMathSciNetGoogle Scholar
  31. [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. MATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    Wilmiam J. Mitchell. Beginning inner model theory. Chapter 17 in this Handbook. 10.1007/978-1-4020-5764-9_18.
  33. [33]
    William J. Mitchell. Ramsey cardinals and constructibility. Journal of Symbolic Logic, 44(2):260–266, 1979. MATHCrossRefMathSciNetGoogle Scholar
  34. [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. [35]
    William J. Mitchell. The core model for sequences of measures. I. Mathematical Proceedings of the Cambridge Philosophical Society, 95(2):229–260, 1984. MATHCrossRefMathSciNetGoogle Scholar
  36. [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. [37]
    William J. Mitchell. Applications of the covering lemma for sequences of measures. Transactions of the American Mathematical Society, 299(1):41–58, 1987. MATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    William J. Mitchell. Definable singularity. Transactions of the American Mathematical Society, 327(1):407–426, 1991. MATHCrossRefMathSciNetGoogle Scholar
  39. [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. [40]
    William J. Mitchell. On the singular cardinal hypothesis. Transactions of the American Mathematical Society, 329(2):507–530, 1992. MATHCrossRefMathSciNetGoogle Scholar
  41. [41]
    William J. Mitchell. Jónsson cardinals, Erdős cardinals, and the core model. Preliminary, available on Logic Eprints, 1994. Google Scholar
  42. [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. [43]
    William J. Mitchell and Ernest Schimmerling. Weak covering without countable closure. Mathematical Research Letters, 2(5):595–609, 1995. MATHMathSciNetGoogle Scholar
  44. [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. MATHCrossRefMathSciNetGoogle Scholar
  45. [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. MATHCrossRefMathSciNetGoogle Scholar
  46. [46]
    Karl Prikry. Changing measurable into accessible cardinals. Dissertationes Mathematicae (Rozprawy Mathematycne), 68:359–378, 1971. MathSciNetGoogle Scholar
  47. [47]
    Ernest Schimmerling. A core model toolbox and guide. Chapter 20 in this Handbook. 10.1007/978-1-4020-5764-9_21.
  48. [48]
    Ernest Schimmerling and W. Hugh Woodin. The Jensen covering property. The Journal of Symbolic Logic, 66(4):1505–1523, 2001. MATHCrossRefMathSciNetGoogle Scholar
  49. [49]
    Ralf Schindler. Weak covering and the tree property. Archive for Mathematical Logic, 38(8):515–520, 1999. MATHCrossRefMathSciNetGoogle Scholar
  50. [50]
    Ralf Schindler. The core model for almost linear iterations. Annals of Pure and Applied Logic, 116(1–3):205–272, 2002. MATHCrossRefMathSciNetGoogle Scholar
  51. [51]
    Ralf Schindler. Iterates of the core model. The Journal of Symbolic Logic, 71:241–251, 2006. MATHCrossRefMathSciNetGoogle Scholar
  52. [52]
    Ralf Schindler and Martin Zeman. Fine structure. Chapter 9 in this Handbook. 10.1007/978-1-4020-5764-9_10.
  53. [53]
    Saharon Shelah. Cardinal Arithmetic, volume 29 of Oxford Logic Guides. Oxford University Press, Oxford, 1994. MATHGoogle Scholar
  54. [54]
    Saharon Shelah. Strong covering without squares. Fundamenta Mathematicae, 166(1–2):87–107, 2000. Saharon Shelah’s anniversary issue. MATHMathSciNetGoogle Scholar
  55. [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. MATHCrossRefMathSciNetGoogle Scholar
  56. [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. [57]
    John R. Steel. An outline of inner model theory. Chapter 19 in this Handbook. 10.1007/978-1-4020-5764-9_20.
  58. [58]
    John R. Steel. The Core Model Iterability Problem, volume 8 of Lecture Notes in Logic. Springer, Berlin, 1996. MATHGoogle Scholar
  59. [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. MATHCrossRefMathSciNetGoogle Scholar
  60. [60]
    Claude Sureson. Excursion en mesurabilite. PhD thesis, City University of New York, 1984. Google Scholar
  61. [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. MATHCrossRefMathSciNetGoogle Scholar
  62. [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. [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. MATHCrossRefMathSciNetGoogle Scholar
  64. [64]
    W. Hugh Woodin. The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. de Gruyter, Berlin, 1999. MATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FloridaGainsvilleUSA

Personalised recommendations