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Handbook of Set Theory pp 1595–1684Cite as

An Outline of Inner Model Theory

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Abstract

This paper outlines the basic theory of canonical inner models satisfying large cardinal hypotheses. It begins with the definition of the models, and their fine structural analysis modulo iterability assumptions. It then outlines how to construct canonical inner models, and prove their iterability, in roughly the greatest generality in which it is currently known how to do this. The paper concludes with some applications: genericity iterations, proofs of generic absoluteness, and a proof that the hereditarily ordinal definable sets of L(ℝ) constitute a canonical inner model.

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Bibliography

  1. Alessandro Andretta, Itay Neeman, and John R. Steel. The domestic levels of K c are iterable. Israel Journal of Mathematics, 125:157–201, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  2. Jon Barwise. Admissible Sets and Structures. Perspectives in Mathematical Logic. Springer, Berlin, 1975.

    MATH  Google Scholar 

  3. Howard S. Becker. Thin collections of sets of projective ordinals and analogs of L. Annals of Mathematical Logic, 19(3):205–241, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  4. Howard S. Becker and Alexander S. Kechris. Sets of ordinals constructible from trees and the third Victoria Delfino problem. In James E. Baumgartner, Donald A. Martin, and Saharon Shelah, editors, Axiomatic Set Theory (Boulder, CO, 1983), pages 13–29. American Mathematical Society, Providence, 1984.

    Google Scholar 

  5. Anthony J. Dodd. The Core Model. Cambridge University Press, Cambridge, 1982.

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. 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 

  8. 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 

  9. Matthew Foreman, Menachem Magidor, and Saharon Shelah. Martin’s Maximum, saturated ideals, and non-regular ultrafilters. Part I. Annals of Mathematics, 127:1–47, 1988.

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. Ronald B. Jensen. Inner models and large cardinals. The Bulletin of Symbolic Logic, 1(4):393–407, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  12. Alexander S. Kechris, Donald A. Martin, and Robert M. Solovay. Introduction to Q-theory. In Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis, editors, Cabal Seminar 79–81, pages 199–282. Springer, Berlin, 1983.

    Chapter  Google Scholar 

  13. Peter Koellner and W. Hugh Woodin. Large cardinals from determinacy. Chapter 23 in this Handbook. 10.1007/978-1-4020-5764-9_24.

  14. Kenneth Kunen. Some applications of iterated ultrapowers in set theory. Annals of Mathematical Logic, 1:179–227, 1970.

    Article  MATH  MathSciNet  Google Scholar 

  15. Benedikt Löwe and John R. Steel. An introduction to core model theory. In S. Barry Cooper and John K. Truss, editors, Sets and Proofs, Logic Colloquium ’97, volume 258 of London Mathematical Society Lecture Note Series, pages 103–157. Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  16. Donald A. Martin and John R. Steel. The extent of scales in L(ℝ). In Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis, editors, Cabal Seminar 79–81, volume 1019 of Lecture Notes in Mathematics, pages 86–96. Springer, Berlin, 1983.

    Chapter  Google Scholar 

  17. Donald A. Martin and John R. Steel. A proof of projective determinacy. Journal of the American Mathematical Society, 2(1):71–125, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  18. Donald A. Martin and John R. Steel. Iteration trees. Journal of the American Mathematical Society, 7(1):1–73, 1994.

    Article  MATH  MathSciNet  Google Scholar 

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

  20. William J. Mitchell. The covering lemma. Chapter 18 in this Handbook. 10.1007/978-1-4020-5764-9_19.

  21. William J. Mitchell. Sets constructible from sequences of ultrafilters. The Journal of Symbolic Logic, 39:57–66, 1974.

    Article  MATH  MathSciNet  Google Scholar 

  22. William J. Mitchell. Hypermeasurable cardinals. In Maurice Boffa, Dirk van Dalen, and Kenneth McAloon, editors, Logic Colloquium ’78 (Mons, 1978), pages 303–316. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1979.

    Google Scholar 

  23. 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 

  24. William J. Mitchell. The core model up to a Woodin cardinal. In Logic, Methodology and Philosophy of Science, IX (Uppsala, 1991), volume 134 of Studies in Logic and the Foundations of Mathematics, pages 157–175. North-Holland, Amsterdam, 1994.

    Chapter  Google Scholar 

  25. William J. Mitchell and John R. Steel. Fine Structure and Iteration Trees, volume 3 of Lecture Notes in Logic. Springer, Berlin, 1994.

    MATH  Google Scholar 

  26. 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 

  27. Yiannis N. Moschovakis. Descriptive Set Theory. North-Holland, Amsterdam, 1980.

    MATH  Google Scholar 

  28. Itay Neeman. Determinacy in L(ℝ). Chapter 22 in this Handbook. 10.1007/978-1-4020-5764-9_23.

  29. Itay Neeman. Optimal proofs of determinacy. The Bulletin of Symbolic Logic, 1(3):327–339, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  30. Itay Neeman. Inner models in the region of a Woodin limit of Woodin cardinals. Annals of Pure and Applied Logic, 116:67–155, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  31. Itay Neeman and John R. Steel. A weak Dodd-Jensen lemma. The Journal of Symbolic Logic, 64(3):1285–1294, 1999.

    Article  MATH  MathSciNet  Google Scholar 

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

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

    Article  MATH  MathSciNet  Google Scholar 

  34. Ernest Schimmerling. A finite family weak square principle. The Journal of Symbolic Logic, 64:1087–1100, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  35. Ernest Schimmerling and John R. Steel. Fine structure for tame inner models. The Journal of Symbolic Logic, 61(2):621–639, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  36. Ernest Schimmerling and John R. Steel. The maximality of the core model. Transactions of the American Mathematical Society, 351:3119–3141, 1999.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

  39. Ralf Schindler, John R. Steel, and Martin Zeman. Deconstructing inner model theory. The Journal of Symbolic Logic, 67:721–736, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  40. John R. Steel. Inner models with many Woodin cardinals. Annals of Pure and Applied Logic, 65(2):185–209, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  41. John R. Steel. HODL(ℝ) is a core model below Θ. The Bulletin of Symbolic Logic, 1(1):75–84, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  42. John R. Steel. Projectively well-ordered inner models. Annals of Pure and Applied Logic, 74(1):77–104, 1995.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  44. John R. Steel. Mathematics needs new axioms. The Bulletin of Symbolic Logic, 6:422–433, 2000.

    MathSciNet  Google Scholar 

  45. John R. Steel. A stationary-tower-free proof of the derived model theorem. In Su Gao, Steve Jackson, and Yi Zhang, editors, Advances in Logic, volume 425 of Contemporary Mathematics, pages 1–8. American Mathematical Society, Providence, 2007.

    Google Scholar 

  46. John R. Steel. Derived models associated to mice. In C. T. Chong, Qi Feng, Theodore Slaman, W. Hugh Woodin, and Y. Yang, editors, Computational Prospects of Infinity, pages 105–193. World Scientific, Singapore, 2008.

    Google Scholar 

  47. Philip D. Welch. Σ* fine structure. Chapter 10 in this Handbook. 10.1007/978-1-4020-5764-9_11.

  48. Philip D. Welch. Combinatorial principles in the core model. PhD dissertation, Oxford, 1979.

    Google Scholar 

  49. Dorshka Wylie. Condensation and square in a higher core model. PhD dissertation, MIT, 1990.

    Google Scholar 

  50. Martin Zeman. Inner Models and Large Cardinals, volume 5 of de Gruyter Series in Logic and Applications. de Gruyter, Berlin, 2002.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of California at Berkeley, Berkeley, CA, 94720, USA

    John R. Steel

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  1. John R. Steel
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Correspondence to John R. Steel .

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Editors and Affiliations

  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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Steel, J.R. (2010). An Outline of Inner Model Theory. In: Foreman, M., Kanamori, A. (eds) Handbook of Set Theory. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5764-9_20

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