Weakly Measurable Cardinals and Partial Near Supercompactness
Item
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Title
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Weakly Measurable Cardinals and Partial Near Supercompactness
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Identifier
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d_2009_2013:1df03b72be31:10934
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identifier
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11196
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Creator
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Schanker, Jason Aaron,
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Contributor
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Joel David Hamkins
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Date
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2011
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics | Applied mathematics
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Abstract
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I will introduce a few new large cardinal concepts. A weakly measurable cardinal is a new large cardinal concept obtained by weakening the familiar concept of a measurable cardinal. Specifically, a cardinal kappa is weakly measurable if for every collection A containing at most kappa+ many subsets of kappa, there exists a nonprincipal kappa-complete filter on kappa measuring all sets in A . Every measurable cardinal is weakly measurable, but a weakly measurable cardinal need not be measurable. Moreover, while the GCH cannot fail first at a measurable cardinal, I will show that it can fail first at a weakly measurable cardinal. More generally, if kappa is measurable, then we can make its weak measurability indestructible by the forcing Add(kappa, eta) for all eta while forcing the GCH to hold below kappa. Nevertheless, I shall prove that weakly measurable cardinals and measurable cardinals are equiconsistent.;A cardinal kappa is nearly theta-supercompact if for every A ⊆ theta, there exists a transitive M |= ZFC- closed under <kappa sequences with A, kappa, theta ∈ M, transitive N, and an elementary embedding j : M → N with critical point kappa such that j(kappa) > theta and j"theta ∈ N. This concept strictly refines the theta-supercompactness hierarchy as every theta-supercompact cardinal is nearly theta-supercompact, and every nearly 2q<k -supercompact cardinal kappa is theta-supercompact. Moreover, if kappa is a theta-supercompact cardinal for some theta such that theta <kappa = theta, we can move to a forcing extension preserving all cardinals below theta++ where kappa remains theta-supercompact but is not nearly theta+-supercompact. I will also show that if kappa is nearly theta-supercompact for some theta ≥ 2 kappa such that theta<theta = theta, then there exists a forcing extension preserving all cardinals at or above kappa where kappa is nearly theta-supercompact but not measurable. These types of large cardinals also come equipped with a nontrivial indestructibility result, and I will prove that if kappa is nearly theta-supercompact for some theta ≥ kappa such that theta<theta = theta, then there is a forcing extension where its near theta-supercompactness is preserved and indestructible by any further <kappa-directed closed theta-c.c. forcing of size at most theta. Finally, these cardinals have high consistency strength. Specifically, I will show that if kappa is nearly kappa +-supercompact for some theta ≥ kappa+ for which theta<theta = theta, then AD holds in L( R ). In particular, if kappa is nearly kappa+-supercompact and 2kappa = kappa+, then AD holds in L( R ).
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Type
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dissertation
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Source
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2009_2013.csv
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degree
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Ph.D.
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Program
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Mathematics
