The Weak Choice Principle WISC may Fail in the Category of Sets
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 ToposesPreview
Unable to display preview. Download preview PDF.
References
- 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.Blass A: Cohomology detects failures of the axiom of choice. Transactions of American Mathematical Society 279(1), 257–269 (1983)CrossRefGoogle Scholar
- 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.Gitik, M., All uncountable cardinals can be singular, Israel Journal of Mathematics 35(1-2):61–88, 1980.Google Scholar
- 5.Karagila A.: Embedding orders into cardinals with \({DC_\kappa}\) . Fundamenta Mathematicae. 226, 143–156 (2014) arXiv:1212.4396.CrossRefGoogle Scholar
- 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.MacLane S., Moerdijk I.: Sheaves in Geometry and Logic,. Springer, Berlin (1992)CrossRefGoogle Scholar
- 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.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.Shulman, M., Stack semantics and the comparison of material and structural set theories, 2010. arXiv:1004.3802.
- 11.van den Berg, B., Predicative toposes, 2012. arXiv:1207.0959.
- 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