Studia Logica

, Volume 103, Issue 5, pp 1005–1017 | Cite as

The Weak Choice Principle WISC may Fail in the Category of Sets

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Abstract

The set-theoretic axiom WISC states that for every set there is a set of surjections to it cofinal in all such surjections. By constructing an unbounded topos over the category of sets and using an extension of the internal logic of a topos due to Shulman, we show that WISC is independent of the rest of the axioms of the set theory given by a well-pointed topos. This also gives an example of a topos that is not a predicative topos as defined by van den Berg.

Keywords

WISC Choice principle Set theory ETCS Toposes 

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References

  1. 1.
    Aczel, P., The type theoretic interpretation of constructive set theory, in Logic Colloquium '77, vol. 96 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1978, pp. 55–66.Google Scholar
  2. 2.
    Blass A: Cohomology detects failures of the axiom of choice. Transactions of American Mathematical Society 279(1), 257–269 (1983)CrossRefGoogle Scholar
  3. 3.
    Dorais, F. G., http://mathoverflow.net/users/2000, On a weak choice principle, MathOverflow, 2012. http://mathoverflow.net/a/99934/ (version: 2012-06-18).
  4. 4.
    Gitik, M., All uncountable cardinals can be singular, Israel Journal of Mathematics 35(1-2):61–88, 1980.Google Scholar
  5. 5.
    Karagila A.: Embedding orders into cardinals with \({DC_\kappa}\) . Fundamenta Mathematicae. 226, 143–156 (2014) arXiv:1212.4396.CrossRefGoogle Scholar
  6. 6.
    Lawvere, F. W., An elementary theory of the category of sets (long version) with commentary, Reprints in Theory and Applications of Categories 11:1–35, 2005. Reprinted and expanded from Proceedings of the National Academy of Sciences of United States of America 5(2), 1964, With comments by the author and Colin McLarty.Google Scholar
  7. 7.
    MacLane S., Moerdijk I.: Sheaves in Geometry and Logic,. Springer, Berlin (1992)CrossRefGoogle Scholar
  8. 8.
    Roberts D.M.: Internal categories, anafunctors and localisation. Theory and Applications of Categories 26(29), 788–829 (2012)arXiv:1101.2363.Google Scholar
  9. 9.
    SGA4—Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, vol. 269, Springer, Berlin, 1972. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier.Google Scholar
  10. 10.
    Shulman, M., Stack semantics and the comparison of material and structural set theories, 2010. arXiv:1004.3802.
  11. 11.
    van den Berg, B., Predicative toposes, 2012. arXiv:1207.0959.
  12. 12.
    van den Berg, B., and I. Moerdijk, The axiom of multiple choice and models for constructive set theory, Journal of Mathematical Logic 14:1, 2014. arXiv:1204.4045.

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesThe University of AdelaideAdelaideAustralia

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