Sandwiching the Consistency Strength of Two Global Choiceless Cardinal Patterns

Arthur W. Apter

Bulletin of the Polish Academy of Sciences. Mathematics (2009)

  • Volume: 57, Issue: 3, page 189-197
  • ISSN: 0239-7269

Abstract

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We provide upper and lower bounds in consistency strength for the theories “ZF + ¬ A C ω + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω” and “ZF + ¬ A C ω + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω₁”. In particular, our models for both of these theories satisfy “ZF + ¬ A C ω + κ is singular iff κ is either an uncountable limit cardinal or the successor of an uncountable limit cardinal”.

How to cite

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Arthur W. Apter. "Sandwiching the Consistency Strength of Two Global Choiceless Cardinal Patterns." Bulletin of the Polish Academy of Sciences. Mathematics 57.3 (2009): 189-197. <http://eudml.org/doc/281299>.

@article{ArthurW2009,
abstract = {We provide upper and lower bounds in consistency strength for the theories “ZF + $¬AC_ω$ + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω” and “ZF + $¬AC_ω$ + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω₁”. In particular, our models for both of these theories satisfy “ZF + $¬AC_ω$ + κ is singular iff κ is either an uncountable limit cardinal or the successor of an uncountable limit cardinal”.},
author = {Arthur W. Apter},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {consistency strength; supercompact cardinal; supercompact Radin forcing; symmetric inner model},
language = {eng},
number = {3},
pages = {189-197},
title = {Sandwiching the Consistency Strength of Two Global Choiceless Cardinal Patterns},
url = {http://eudml.org/doc/281299},
volume = {57},
year = {2009},
}

TY - JOUR
AU - Arthur W. Apter
TI - Sandwiching the Consistency Strength of Two Global Choiceless Cardinal Patterns
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2009
VL - 57
IS - 3
SP - 189
EP - 197
AB - We provide upper and lower bounds in consistency strength for the theories “ZF + $¬AC_ω$ + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω” and “ZF + $¬AC_ω$ + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω₁”. In particular, our models for both of these theories satisfy “ZF + $¬AC_ω$ + κ is singular iff κ is either an uncountable limit cardinal or the successor of an uncountable limit cardinal”.
LA - eng
KW - consistency strength; supercompact cardinal; supercompact Radin forcing; symmetric inner model
UR - http://eudml.org/doc/281299
ER -

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