The consistency strength of the tree property at the double successor of a measurable cardina

Natasha Dobrinen; Sy-David Friedman

Fundamenta Mathematicae (2010)

  • Volume: 208, Issue: 2, page 123-153
  • ISSN: 0016-2736

Abstract

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The Main Theorem is the equiconsistency of the following two statements: (1) κ is a measurable cardinal and the tree property holds at κ⁺⁺; (2) κ is a weakly compact hypermeasurable cardinal. From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable cardinal and a measurable cardinal far enough above it, then there is an inner model in which there is a proper class of measurable cardinals, and in which the tree property holds at the double successor of each strongly inaccessible cardinal. If 0 exists, then we can construct an inner model in which the tree property holds at the double successor of each strongly inaccessible cardinal. We also find upper and lower bounds for the consistency strength of there being no special Aronszajn trees at the double successor of a measurable cardinal. The upper and lower bounds differ only by 1 in the Mitchell order.

How to cite

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Natasha Dobrinen, and Sy-David Friedman. "The consistency strength of the tree property at the double successor of a measurable cardina." Fundamenta Mathematicae 208.2 (2010): 123-153. <http://eudml.org/doc/283162>.

@article{NatashaDobrinen2010,
abstract = {The Main Theorem is the equiconsistency of the following two statements: (1) κ is a measurable cardinal and the tree property holds at κ⁺⁺; (2) κ is a weakly compact hypermeasurable cardinal. From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable cardinal and a measurable cardinal far enough above it, then there is an inner model in which there is a proper class of measurable cardinals, and in which the tree property holds at the double successor of each strongly inaccessible cardinal. If $0^\{#\}$ exists, then we can construct an inner model in which the tree property holds at the double successor of each strongly inaccessible cardinal. We also find upper and lower bounds for the consistency strength of there being no special Aronszajn trees at the double successor of a measurable cardinal. The upper and lower bounds differ only by 1 in the Mitchell order.},
author = {Natasha Dobrinen, Sy-David Friedman},
journal = {Fundamenta Mathematicae},
keywords = {Sacks forcing; measurable cardinal; tree property; hypermeasurable cardinal; consistency; inner model},
language = {eng},
number = {2},
pages = {123-153},
title = {The consistency strength of the tree property at the double successor of a measurable cardina},
url = {http://eudml.org/doc/283162},
volume = {208},
year = {2010},
}

TY - JOUR
AU - Natasha Dobrinen
AU - Sy-David Friedman
TI - The consistency strength of the tree property at the double successor of a measurable cardina
JO - Fundamenta Mathematicae
PY - 2010
VL - 208
IS - 2
SP - 123
EP - 153
AB - The Main Theorem is the equiconsistency of the following two statements: (1) κ is a measurable cardinal and the tree property holds at κ⁺⁺; (2) κ is a weakly compact hypermeasurable cardinal. From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable cardinal and a measurable cardinal far enough above it, then there is an inner model in which there is a proper class of measurable cardinals, and in which the tree property holds at the double successor of each strongly inaccessible cardinal. If $0^{#}$ exists, then we can construct an inner model in which the tree property holds at the double successor of each strongly inaccessible cardinal. We also find upper and lower bounds for the consistency strength of there being no special Aronszajn trees at the double successor of a measurable cardinal. The upper and lower bounds differ only by 1 in the Mitchell order.
LA - eng
KW - Sacks forcing; measurable cardinal; tree property; hypermeasurable cardinal; consistency; inner model
UR - http://eudml.org/doc/283162
ER -

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