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Choice principles in hyperuniverses

Marco Forti & Furio Honsell

Annals of Pure and Applied Logic 77 (1):35-52 (1996)

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On the Consistency of a Positive Theory.Olivier Esser - 1999 - Mathematical Logic Quarterly 45 (1):105-116.details In positive theories, we have an axiom scheme of comprehension for positive formulas. We study here the “generalized positive” theory GPK∞+. Natural models of this theory are hyperuniverses. The author has shown in [2] that GPK∞+ interprets the Kelley Morse class theory. Here we prove that GPK∞+ + ACWF and the Kelley-Morse class theory with the axiom of global choice and the axiom “On is ramifiable” are mutually interpretable. This shows that GPK∞+ + ACWF is a “strong” theory since “On (...) Download Export citation Bookmark 14 citations
Inconsistency of the Axiom of Choice with the Positive Theory $GPK^+ \infty$.Olivier Esser - 2000 - Journal of Symbolic Logic 65 (4):1911-1916.details The idea of the positive theory is to avoid the Russell's paradox by postulating an axiom scheme of comprehension for formulas without "too much" negations. In this paper, we show that the axiom of choice is inconsistent with the positive theory $GPK^+ \infty$. Download Export citation Bookmark 2 citations
(15 other versions)2000 European Summer Meeting of the Association for Symbolic Logic. Logic Colloquium 2000.Carol Wood - 2001 - Bulletin of Symbolic Logic 7 (1):82-163.details Download Export citation Bookmark
About the coexistence of “classical sets” with “non-classical” ones: A survey.Roland Hinnion - 2003 - Logic and Logical Philosophy 11:79-90.details This is a survey of some possible extensions of ZF to a larger universe, closer to the “naive set theory” (the universes discussed here concern, roughly speaking : stratified sets, partial sets, positive sets, paradoxical sets and double sets). Download Export citation Bookmark 2 citations
Addendum and corrigendum Choice Principles in Hyperuniverses Annals of Pure and Applied Logic 77 (1996) 35–52.Marco Forti & Furio Honsell - 1998 - Annals of Pure and Applied Logic 92 (2):211-214.details The proof of Lemma 5 in our paper “Choice Principles in Hyperuniverses” [3], contains an error. In the present note we show that the statement of that lemma is false and hence the Axiom of Choice fails in all κ-hyperuniverses, for uncountable κ. However, a weaker version of Lemma 5 can be proved, which implies that the Linear Ordering Principle holds in all κ-metric κ-hyperuniverses. Download Export citation Bookmark 2 citations

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