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H 1 The hierarchy of second-order set theories between GBC and KM and beyond

Auteurs : Hamkins, Joel David (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : Recent work has clarified how various natural second-order set-theoretic principles, such as those concerned with class forcing or with proper class games, fit into a new robust hierarchy of second-order set theories between Gödel-Bernays GBC set theory and Kelley-Morse KM set theory and beyond. For example, the principle of clopen determinacy for proper class games is exactly equivalent to the principle of elementary transfinite recursion ETR, strictly between GBC and GBC+$\Pi^1_1$-comprehension; open determinacy for class games, in contrast, is strictly stronger; meanwhile, the class forcing theorem, asserting that every class forcing notion admits corresponding forcing relations, is strictly weaker, and is exactly equivalent to the fragment $\text{ETR}_{\text{Ord}}$ and to numerous other natural principles. What is emerging is a higher set-theoretic analogue of the familiar reverse mathematics of second-order number theory.

    Codes MSC :
    03C62 - Models of arithmetic and set theory
    03E30 - Axiomatics of classical set theory and its fragments
    03E60 - Determinacy principles

    Ressources complémentaires :

    Informations sur la rencontre

    Nom du congrès : 14th International workshop in set theory / XIVe Atelier international de théorie des ensembles
    Organisteurs Congrès : Dzamonja, Mirna ; Magidor, Menachem ; Velickovic, Boban ; Woodin, W. Hugh
    Dates : 09/10/2017 - 13/10/2017
    Année de la rencontre : 2017
    URL Congrès : http://conferences.cirm-math.fr/1606.html

    Citation Data

    DOI : 10.24350/CIRM.V.19228403
    Cite this video as: Hamkins, Joel David (2017). The hierarchy of second-order set theories between GBC and KM and beyond. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19228403
    URI : http://dx.doi.org/10.24350/CIRM.V.19228403

    Voir aussi


    1. Gitman, V., & Hamkins, J.D. (2017). Open determinacy for class games. In A.E. Caicedo, J. Cummings, & P. Koellner (Eds.), Foundations of mathematics (pp. 121-143). Providence, RI: American Mathematical Society - http://dx.doi.org/10.1090/conm/690/13865

    2. Gitman, V., Hamkins, J.D., Holy, P., Schlicht, P., & Williams, K. (2017). The exact strength of the class forcing theorem. - https://arxiv.org/abs/1707.03700

    3. Gitman, V., Hamkins, J.D., & Johnstone, T. Kelley-Morse set theory and choice principles for classes. Manuscript in preparation -