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Indestructibility, measurability, and degrees of supercompactness
Author(s) -
Apter Arthur W.
Publication year - 2012
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201020097
Subject(s) - mathematics , limit (mathematics) , forcing (mathematics) , regular cardinal , distributive property , pure mathematics , combinatorics , mathematical physics , mathematical analysis
Suppose that κ is indestructibly supercompact and there is a measurable cardinal λ > κ. It then follows that A 1 = {δ < κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is not δ + supercompact} is unbounded in κ. If in addition λ is 2 λ supercompact, then A 2 = {δ < κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is δ + supercompact} is unbounded in κ as well. The large cardinal hypotheses on λ are necessary, as we further demonstrate by constructing via forcing two distinct models in which either \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$A_1 = \varnothing$\end{document} or \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$A_2 = \varnothing$\end{document} . In each of these models, there is an indestructibly supercompact cardinal κ, and a restricted large cardinal structure above κ. If we weaken the indestructibility requirement on κ to indestructibility under partial orderings which are both κ‐directed closed and (κ + , ∞)‐distributive, then it is possible to construct a model containing a supercompact cardinal κ witnessing this degree of indestructibility in which every measurable cardinal δ < κ is (at least) δ + supercompact.