A Note on Indestructibility and Strong Compactness

Arthur W. Apter

Bulletin of the Polish Academy of Sciences. Mathematics (2008)

  • Volume: 56, Issue: 3, page 191-197
  • ISSN: 0239-7269

Abstract

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If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ⁺,∞)-distributive and λ is 2 λ supercompact, then by a result of Apter and Hamkins [J. Symbolic Logic 67 (2002)], δ < κ | δ is δ⁺ strongly compact yet δ is not δ⁺ supercompact must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in which no cardinal δ > κ is 2 δ = δ ⁺ supercompact, κ’s supercompactness is indestructible under κ-directed closed forcing which is also (κ⁺,∞)-distributive, and for every measurable cardinal δ, δ is δ⁺ strongly compact iff δ is δ⁺ supercompact.

How to cite

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Arthur W. Apter. "A Note on Indestructibility and Strong Compactness." Bulletin of the Polish Academy of Sciences. Mathematics 56.3 (2008): 191-197. <http://eudml.org/doc/281138>.

@article{ArthurW2008,
abstract = {If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ⁺,∞)-distributive and λ is $2^λ$ supercompact, then by a result of Apter and Hamkins [J. Symbolic Logic 67 (2002)], δ < κ | δ is δ⁺ strongly compact yet δ is not δ⁺ supercompact must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in which no cardinal δ > κ is $2^δ = δ⁺$ supercompact, κ’s supercompactness is indestructible under κ-directed closed forcing which is also (κ⁺,∞)-distributive, and for every measurable cardinal δ, δ is δ⁺ strongly compact iff δ is δ⁺ supercompact.},
author = {Arthur W. Apter},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {supercompact; strongly compact; indestructibility; nonreflecting stationary set of ordinals},
language = {eng},
number = {3},
pages = {191-197},
title = {A Note on Indestructibility and Strong Compactness},
url = {http://eudml.org/doc/281138},
volume = {56},
year = {2008},
}

TY - JOUR
AU - Arthur W. Apter
TI - A Note on Indestructibility and Strong Compactness
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2008
VL - 56
IS - 3
SP - 191
EP - 197
AB - If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ⁺,∞)-distributive and λ is $2^λ$ supercompact, then by a result of Apter and Hamkins [J. Symbolic Logic 67 (2002)], δ < κ | δ is δ⁺ strongly compact yet δ is not δ⁺ supercompact must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in which no cardinal δ > κ is $2^δ = δ⁺$ supercompact, κ’s supercompactness is indestructible under κ-directed closed forcing which is also (κ⁺,∞)-distributive, and for every measurable cardinal δ, δ is δ⁺ strongly compact iff δ is δ⁺ supercompact.
LA - eng
KW - supercompact; strongly compact; indestructibility; nonreflecting stationary set of ordinals
UR - http://eudml.org/doc/281138
ER -

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