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The linearity of the Mitchell order

Gabriel Goldberg

Journal of Mathematical Logic 18 (1):1850005 (2018)

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Small models, large cardinals, and induced ideals.Peter Holy & Philipp Lücke - 2021 - Annals of Pure and Applied Logic 172 (2):102889.details We show that many large cardinal notions up to measurability can be characterized through the existence of certain filters for small models of set theory. This correspondence will allow us to obtain a canonical way in which to assign ideals to many large cardinal notions. This assignment coincides with classical large cardinal ideals whenever such ideals had been defined before. Moreover, in many important cases, relations between these ideals reflect the ordering of the corresponding large cardinal properties both under direct (...) Download Export citation Bookmark 3 citations
Indestructibility and the linearity of the Mitchell ordering.Arthur W. Apter - 2024 - Archive for Mathematical Logic 63 (3):473-482.details Suppose that \(\kappa \) is indestructibly supercompact and there is a measurable cardinal \(\lambda > \kappa \). It then follows that \(A_0 = \{\delta is a measurable cardinal and the Mitchell ordering of normal measures over \(\delta \) is nonlinear \(\}\) is unbounded in \(\kappa \). If the Mitchell ordering of normal measures over \(\lambda \) is also linear, then by reflection (and without any use of indestructibility), \(A_1= \{\delta is a measurable cardinal and the Mitchell ordering of normal measures (...) Download Export citation Bookmark
Controlling the number of normal measures at successor cardinals.Arthur W. Apter - 2022 - Mathematical Logic Quarterly 68 (3):304-309.details We examine the number of normal measures a successor cardinal can carry, in universes in which the Axiom of Choice is false. When considering successors of singular cardinals, we establish relative consistency results assuming instances of supercompactness, together with the Ultrapower Axiom (introduced by Goldberg in [12]). When considering successors of regular cardinals, we establish relative consistency results only assuming the existence of one measurable cardinal. This allows for equiconsistencies. Download Export citation Bookmark

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