Easton’s theorem and large cardinals

Annals of Pure and Applied Logic 154 (3):191-208 (2008)
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Abstract

The continuum function αmaps to2α on regular cardinals is known to have great freedom. Let us say that F is an Easton function iff for regular cardinals α and β, image and α<β→F≤F. The classic example of an Easton function is the continuum function αmaps to2α on regular cardinals. If GCH holds then any Easton function is the continuum function on regular cardinals of some cofinality-preserving extension V[G]; we say that F is realised in V[G]. However if we also wish to preserve measurable cardinals, new restrictions must be put on F. We say that κ is F-hypermeasurable iff there is an elementary embedding j:V→M with critical point κ such that H)Vsubset of or equal toM; j will be called a witnessing embedding. We will show that if GCH holds then for any Easton function F there is a cofinality-preserving generic extension V[G] such that if κ, closed under F, is F-hypermeasurable in V and there is a witnessing embedding j such that j≥F, then κ will remain measurable in V[G]

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Citations of this work

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Easton's theorem for Ramsey and strongly Ramsey cardinals.Brent Cody & Victoria Gitman - 2015 - Annals of Pure and Applied Logic 166 (9):934-952.
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On supercompactness and the continuum function.Brent Cody & Menachem Magidor - 2014 - Annals of Pure and Applied Logic 165 (2):620-630.

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References found in this work

[Omnibus Review].Akihiro Kanamori - 1981 - Journal of Symbolic Logic 46 (4):864-866.
The negation of the singular cardinal hypothesis from o(K)=K++.Moti Gitik - 1989 - Annals of Pure and Applied Logic 43 (3):209-234.
Perfect-set forcing for uncountable cardinals.Akihiro Kanamori - 1980 - Annals of Mathematical Logic 19 (1-2):97-114.
Perfect trees and elementary embeddings.Sy-David Friedman & Katherine Thompson - 2008 - Journal of Symbolic Logic 73 (3):906-918.

View all 6 references / Add more references