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- Publisher
- Springer Journals
- Copyright
- Copyright © 2011 by Springer-Verlag
- Subject
- Mathematics; Mathematics, general; Algebra; Mathematical Logic and Foundations
- ISSN
- 0933-5846
- eISSN
- 1432-0665
- DOI
- 10.1007/s00153-011-0233-z
- Publisher site
- See Article on Publisher Site
Abstract
We show that the consistency of the theory “ZF + DC + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” follows from the consistency of a proper class of supercompact cardinals. This extends earlier results due to the author showing that the consistency of the theory “ $${{\rm ZF} + \neg{\rm AC}_\omega}$$ + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” follows from hypotheses stronger in consistency strength than a supercompact limit of supercompact cardinals. A lower bound in consistency strength is provided by a result of Busche and Schindler, who showed that the consistency of the theory “ZF + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” implies the consistency of AD L(R).
Journal
Archive for Mathematical Logic – Springer Journals
Published: Apr 1, 2011