Skip to main content
SpringerLink
Log in

Beginning Inner Model Theory

  • Chapter
  • First Online:
Handbook of Set Theory

Abstract

This chapter provides an introduction to the basic theory of inner models of set theory, without fine structure. Section 1 begins with the basic theory of Gödel’s class L of constructible sets, with an emphasis on the condensation property, introduces sharps, and includes a brief discussion of the Dodd-Jensen core model. The next two sections describe the extension of these concepts to arbitrary sequences of measures, and then via extender models to cardinal properties stronger than measurability. Section 4 gives a summary of the status and known properties of inner models for cardinals ranging from strong to supercompact, and the final section discusses core models.

This research was partly supported by NSF grant DPS-9970536. The author would like to thank John Krueger, Paul Larson, Diego Rojas, and the referee for valuable corrections and suggestions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 709.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 899.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 899.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Aaron Beller, Ronald B. Jensen, and Philip D. Welch. Coding the Universe, volume 47 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1982.

    MATH  Google Scholar 

  3. Keith J. Devlin. Constructibility. Springer, Berlin, 1984.

    MATH  Google Scholar 

  4. Anthony J. Dodd. Core models. The Journal of Symbolic Logic, 48(1):78–90, 1983.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. 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 

  7. 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 

  8. Matthew Foreman, Menachem Magidor, and Saharon Shelah. Martin’s maximum, saturated ideals and nonregular ultrafilters. II. Annals of Mathematics (2), 127(3):521–545, 1988.

    Article  MathSciNet  Google Scholar 

  9. 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 

  10. Sy D. Friedman. Constructibility and class forcing. Chapter 8 in this Handbook. 10.1007/978-1-4020-5764-9_9.

  11. Sy D. Friedman. Provable \({\Pi\sp1\sb2}\) -singletons. Proceedings of the American Mathematical Society, 123(9):2873–2874, 1995.

    Article  MATH  MathSciNet  Google Scholar 

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

  13. Moti Gitik. On closed unbounded sets consisting of former regulars. The Journal of Symbolic Logic, 64(1):1–12, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  14. Thomas J. Jech. Set Theory. Springer Monographs in Mathematics. Springer, Berlin, 2002. Third millennium edition, revised and expanded.

    Google Scholar 

  15. Akihiro Kanamori. The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Springer, Berlin, 1994.

    MATH  Google Scholar 

  16. Akihiro Kanamori. The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Springer, Berlin, 2003. Second edition.

    MATH  Google Scholar 

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

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. 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 

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

    MATH  MathSciNet  Google Scholar 

  22. 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 

  23. Adrian R. D. Mathias. On sequences generic in the sense of Prikry. Journal of the Australian Mathematical Society, 15:409–414, 1973.

    Article  MATH  MathSciNet  Google Scholar 

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

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

    Article  MATH  MathSciNet  Google Scholar 

  26. 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 

  27. William J. Mitchell. The core model up to a Woodin cardinal. In Dag Prawitz, Brian Skyrms, Dag Westerstahl, editors, 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 

  28. William J. Mitchell. A measurable cardinal with a closed unbounded set of inaccessibles from o(κ)=κ. Transactions of the American Mathematical Society, 353(12):4863–4897, 2001.

    Article  MATH  MathSciNet  Google Scholar 

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

  30. Itay Neeman. Inner models in the region of a Woodin limit of Woodin cardinals. Annals of Pure and Applied Logic, 116(1–3):67–155, 2002.

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  32. Lon B. Radin. Adding closed cofinal sequences to large cardinals. Annals of Mathematical Logic, 22(3):243–261, 1982.

    Article  MATH  MathSciNet  Google Scholar 

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

  34. 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 

  35. 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 

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

  37. Dana S. Scott. Measurable cardinals and constructible sets. Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, 9:521–524, 1955.

    Google Scholar 

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

  39. John R. Steel. Core models with more Woodin cardinals. The Journal of Symbolic Logic, 67(3):1197–1226, 2002.

    Article  MATH  MathSciNet  Google Scholar 

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

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

    MATH  Google Scholar 

  42. Martin Zeman. Inner Models and Large Cardinals. de Gruyter, Berlin, 2002.

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    William J. Mitchell

Authors
  1. William J. Mitchell
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to William J. Mitchell .

Editor information

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

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Science+Business Media B.V.

About this chapter

Cite this chapter

Mitchell, W.J. (2010). Beginning 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_18

Download citation

Publish with us

Policies and ethics

Search

Navigation