Skip to main content
SpringerLink
Log in

Determinacy implies that ℵ2 is supercompact

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Assuming the axiom of determinacy, ℵ1 and ℵ2 areδ-supercompact, whereδ is a fairly large cardinal which will be defined below.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Becker,Thin collections of sets of projective ordinals and analogs of L, Ann. Math. Logic19 (1980), 205–241.

    Article  MATH  MathSciNet  Google Scholar 

  2. H. Becker, ADand the supercompactness of ℵ1, J. Symbolic Logic46 (1981), to appear.

  3. C. A. DiPrisco and J. Henle,On the compactness of ℵ 1 and ℵ2, J. Symbolic Logic43 (1978), 394–401.

    Article  MathSciNet  Google Scholar 

  4. L. A. Harrington and A. S. Kechris,On the determinacy of games on ordinals, Ann. Math. Logic20 (1981), 109–154.

    Article  MATH  MathSciNet  Google Scholar 

  5. T. J. Jech,Set Theory, Academic Press, New York, 1978.

    Google Scholar 

  6. A. Kanamori and M. Magidor,The evolution of large cardinal axioms in set theory, inHigher Set Theory (G. H. Müller and D. S. Scott, eds.), Lecture Notes in Mathematics Vol. 669, Springer-Verlag, Berlin-Heidelberg-New York, 1978, pp. 99–275.

    Chapter  Google Scholar 

  7. A. S. Kechris and W. H. Woodin,Generic codes for uncountable ordinals, partition properties and elementary embeddings, to appear.

  8. A. S. Kechris, E. M. Kleinberg, Y. N. Moschovakis and W. H. Woodin,The axiom of determinacy, strong partition properties, and non-singular measures, inCabal Seminar 77–79 (A. S. Kechris, D. A. Martin and Y. N. Moschovakis, eds.), Lecture Notes in Mathematics839, Springer-Verlag, Berlin-Heidelberg-New York, 1981, pp. 75–99.

    Chapter  Google Scholar 

  9. D. A. Martin,The axiom of determinateness and reduction principles in the analytical hierarchy, Bull Amer. Math. Soc.74 (1968), 687–689.

    Article  MATH  MathSciNet  Google Scholar 

  10. D. A. Martin, Y. N. Moschovakis and J. R. Steel, to appear.

  11. Y. N. Moschovakis,Determinacy and prewellorderings of the continuum, inMathematical Logic and Foundations of Set Theory (Y. Bar-Hillel, ed.), North-Holland, Amsterdam, 1970, pp. 24–62.

    Google Scholar 

  12. Y. N. Moschovakis,Descriptive Set Theory, North-Holland, Amsterdam, 1980.

    MATH  Google Scholar 

  13. J. C. Oxtoby,Measure and Category, Springer, New York, 1971.

    MATH  Google Scholar 

  14. R. M. Solovay,The independence of DC from AD, inCabal Seminar 76–77 (A. S. Kechris and Y. N. Moschovakis, eds.), Lecture Notes in Mathematics, Vol. 689, Springer-Verlag, Berlin-Heidelberg-New York 1978, pp. 171–184.

    Chapter  Google Scholar 

  15. R. M. Solovay, W. N. Reinhardt and A. Kanamori,Strong axioms of infinity and elementary embeddings, Ann. Math. Logic13 (1978), 73–116.

    Article  MATH  MathSciNet  Google Scholar 

  16. W. H. Woodin, ℵ1-dense ideals, to appear.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin, 53706, Madison, WI, USA

    Howard Becker

Authors
  1. Howard Becker
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Becker, H. Determinacy implies that ℵ2 is supercompact. Israel J. Math. 40, 229–234 (1981). https://doi.org/10.1007/BF02761364

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02761364

Keywords

Search

Navigation