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Cofinality changes required for a large set of unapproachable ordinals below $ \aleph _{\omega +1}$


Author: M. C. Stanley
Journal: Proc. Amer. Math. Soc. 135 (2007), 2619-2622
MSC (2000): Primary 03E05
DOI: https://doi.org/10.1090/S0002-9939-07-08760-6
Published electronically: February 28, 2007
MathSciNet review: 2302583
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Abstract: In $ V$, assume that $ \aleph _{\omega }$ is a strong limit cardinal and $ 2^{\aleph _{\omega }}=\aleph _{\omega +1}$. Let $ A$ be the set of approachable ordinals less than $ \aleph _{\omega +1}$. An open question of M. Foreman is whether $ A$ can be non-stationary in some $ \aleph _{\omega }$ and $ \aleph _{\omega +1}$ preserving extension of $ V$. It is shown here that if $ W$ is such an outer model, then $ {\{\,k<\omega :\mathop{\text{cf}}^{W}(\aleph ^{V}_{k})=\aleph ^{W}_{n}\,\}}$ is infinite, for each positive integer $ n$.


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Additional Information

M. C. Stanley
Affiliation: Mathematics Department, San Jose State University, San Jose, California 95192
Email: stanley@math.sjsu.edu

DOI: https://doi.org/10.1090/S0002-9939-07-08760-6
Keywords: Approachable ordinal, $I[\lambda ]$, cofinality, Erd\H os-Rado
Received by editor(s): December 6, 2005
Received by editor(s) in revised form: April 19, 2006, and April 28, 2006
Published electronically: February 28, 2007
Additional Notes: Research supported by NSF grant DMS-0501114
Communicated by: Julia Knight
Article copyright: © Copyright 2007 American Mathematical Society

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