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# Documents  Hamkins, Joel David | enregistrements trouvés : 2

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## The hierarchy of second-order set theories between GBC and KM and beyond Hamkins, Joel David | CIRM H

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Research talks;Logic and Foundations

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. 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, ...

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## Effective mathematics of the uncountable Greenberg, Noam ; Hamkins, Joel David ; Hirschfeldt, Denis R. ; Miller, Russell | Cambridge University Press;Association for Symbolic Logic 2013

Ouvrage

- viii; 197 p.
ISBN 978-1-107-01451-0

Lecture notes in logic

Localisation : Ouvrage RdC (EFFE)

théorie des modèles # algèbre commutative # algèbre différentielle # théorie des graphes # théorie des ensembles

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