Advertisement

Set Theory pp 399-412 | Cite as

Postscript IV: Landmarks of Modern Set Theory

  • Abhijit Dasgupta
Chapter

Abstract

This part contains brief informal discussions (with proofs and most details omitted) of some of the landmark results of set theory of the past 75 years. Topics discussed are constructibility, forcing and independence results, large cardinal axioms, infinite games and determinacy, projective determinacy, and the status of the Continuum Hypothesis.

References

  1. 1.
    J. Bagaria. Natural axioms of set theory and the continuum problem. In Proceedings of the 12th International Congress of Logic, Methodology, and Philosophy of Science, pages 43–64. King’s College London Publications, 2005.Google Scholar
  2. 2.
    J. L. Bell. Boolean-Valued Models and Independence Proofs in Set Theory, volume 12 of Oxford Logic Guides. Clarendon Press, Oxford, 2nd edition, 1985.Google Scholar
  3. 8.
    P. J. Cohen. Set Theory and the Continuum Hypothesis. WA Benjamin, 1966.zbMATHGoogle Scholar
  4. 14.
    F. R. Drake. Set theory: An Introduction to Large Cardinals. North Holland, 1974.zbMATHGoogle Scholar
  5. 16.
    S. Feferman, H. M. Friedman, P. Maddy, and J. R. Steel. Does mathematics need new axioms? The Bulletin of Symbolic Logic, 6(4):401–446, 2000.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 17.
    M. Foreman. Has the Continuum Hypothesis been settled? Talk presented in Logic Colloquium 2003 (Helsinki).Google Scholar
  7. 18.
    M. Foreman and A. Kanamori. Handbook of Set Theory. Springer, 2010.CrossRefzbMATHGoogle Scholar
  8. 22.
    K. Gödel. The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. Number 3 in Annals of Mathematics Studies. Princeton, 1940. Seventh printing, 1966.Google Scholar
  9. 23.
    K. Gödel. What is Cantor’s continuum problem? In P. Benacerraf and H. Putnam, editors, Philosophy of Mathematics: Selected Readings, pages 470–485. Cambridge University Press, second edition, 1983.Google Scholar
  10. 34.
    T. Jech. Set theory: The Third Millennium Edition. Springer, 2003.Google Scholar
  11. 37.
    A. Kanamori. The Higher Infinite. Springer, 2nd edition, 2003.Google Scholar
  12. 42.
    K. Kunen. Set Theory: An Introduction to Independence Proofs, volume 102 of Studies in Logic and the Foundations of Mathematics. Elsevier, 1980.Google Scholar
  13. 44.
    K. Kunen. Set Theory. College Publications, 2011.zbMATHGoogle Scholar
  14. 50.
    P. Maddy. Believing the axioms. I and II. The Journal of Symbolic Logic, 53(2, 3):481–511, 736–764, 1988.Google Scholar
  15. 51.
    P. Maddy. Defending the axioms: On the philosophical foundations of set theory. Oxford University Press Oxford, 2011.CrossRefGoogle Scholar
  16. 52.
    M. Magidor. Some set theories are more equal. preprint, 201?Google Scholar
  17. 53.
    D. A. Martin and A. S. Kechris. Infinite games and effective descriptive set theory. In Analytic Sets [64], pages 403–470.Google Scholar
  18. 55.
    Y. N. Moschovakis. Descriptive Set Theory, volume 155 of Mathematical Surveys and Monographs. American Mathematical Society, 2009.Google Scholar
  19. 56.
    J. Mycielski. On the axiom of determinateness. Fund. Math, 53:205–224, 1964.MathSciNetzbMATHGoogle Scholar
  20. 57.
    J. Mycielski. Games with perfect information. In Handbook of game theory with economic applications, volume 1, pages 41–70. Elsevier, 1992.Google Scholar
  21. 82.
    W. H. Woodin. The continuum hypothesis, Part I and Part II. Notices of the American Mathematical Society, 48(6, 7):567–576, 681–690, 2001.Google Scholar
  22. 83.
    W. H. Woodin. Set theory after Russell: The journey back to Eden. In Link [49], pages 29–47.Google Scholar
  23. 87.
    P. Koellner. The Continuum Hypothesis. The Stanford Encyclopedia of Philosophy (Summer 2013 Edition), E. N. Zalta (ed.). http://plato.stanford.edu/archives/sum2013/entries/continuum-hypothesis/.

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Detroit MercyDetroitUSA

Personalised recommendations