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manuscripta mathematica

, Volume 38, Issue 3, pp 325–332 | Cite as

The “world's simplest axiom of choice” fails

  • M. P. Fourman
  • A. Ščedrov
Article

Abstract

We use topos-theoretic methods to show that intuitionistic set theory with countable or dependent choice does not imply that every family, all of whose elements are doubletons and which has at most one element, has a choice function.

Keywords

Number Theory Algebraic Geometry Topological Group Choice Function Simple Axiom 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    DUMMETT, M.A.E.: Elements of Intuitionism. Oxford University Press, Oxford (1977)Google Scholar
  2. 2.
    FOURMAN, M.P.: The logic of topoi. In: Handbook of Mathematical Logic (ed. J. Barwise), North Holland, Amsterdam (1977), 1053–1090Google Scholar
  3. 3.
    FOURMAN, M.P.: Sheaf models for set theory. J. Pure and Applied Algebra 19 (1980), 91–101Google Scholar
  4. 4.
    FOURMAN, M.P. and GRAYSON, R.J.: Formal spaces induction principles and completeness theorems. In preparationGoogle Scholar
  5. 5.
    FOURMAN, M.P. and HYLAND, J.M.E.: Sheaf models for analysis. In: Applications of Sheaves (Proc. L.M.S. Durham Symposium, 1977), Springer Lecture Notes in Math. 753 (1979)Google Scholar
  6. 6.
    FOURMAN, M.P. and SCOTT, D.S.: Sheaves and logic. In: Applications of Sheaves (Proc. L.M.S. Durham Symposium, 1977), Springer Lecture Notes in Math. 753 (1979)Google Scholar
  7. 7.
    FREYD, P.: The axiom of choice. J. Pure and Applied Algebra 19 (1981), 103–125Google Scholar
  8. 8.
    JOYAL, A. and REYES, G.: Generic models in categorical logic. J. Pure and Applied Algebra (to appear)Google Scholar
  9. 9.
    KRIPKE, S.: Semantical analysis of intuitionistic logic I. Z. math. logik u Grundl. Math. 9 (1963), 67–96Google Scholar
  10. 10.
    LAWVERE, F.W.: Introduction. In: Toposes, Algebraic Geometry and Logic. Springer Lecture Notes in Math. 274 (1972), 1–12Google Scholar
  11. 11.
    MACLANE, S.: Categories for the Working Mathematician. Springer-Verlag, HeidelbergGoogle Scholar
  12. 12.
    MAKKAI, M. and REYES, G.: First-order categorical logic. Springer Lecture Notes in Math. 611 (1977)Google Scholar
  13. 13.
    OSIUS, G.: Logical and Set-Theoretical Tools in Elementary Topoi. In: Model Theory and Topoi. Springer Lecture Notes in Math. 445 (1975), 297–346Google Scholar
  14. 14.
    ŠČEDROV, A.: Consistency and Independence Proofs in intuitionistic set theory. Proceedings of the New Mexico Conference on Constructive Mathematics (F. Richman, ed.), Springer Lecture Notes (to appear)Google Scholar
  15. 15.
    SCOTT, D.S.: Sheaf models for set theory. To appearGoogle Scholar
  16. 16.
    TIERNEY, M.: Forcing Topologies and Classifying Topoi, in Algebra, Topology and Category Theory; a collection of papers in honor of Samuel Eilenberg (A. Heller, ed.), Academic Press, New York (1976), 211–219Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • M. P. Fourman
    • 1
  • A. Ščedrov
    • 2
  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA

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