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Singularizing successor cardinals by forcing

Authors: Dominik Adolf, Arthur W. Apter and Peter Koepke
Journal: Proc. Amer. Math. Soc. 146 (2018), 773-783
MSC (2010): Primary 03E35, 03E45, 03E55
Published electronically: November 6, 2017
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Abstract: We exhibit models of set theory, using large cardinals and forcing, in which successor cardinals can be made singular by some ``Namba-like'' further set forcing, in most cases without collapsing cardinals below that successor cardinal. For successors of regular cardinals, we work from consistency-wise optimal assumptions in the ground model. Successors of singular cardinals require stronger hypotheses. Our partial orderings are different from Woodin's stationary tower forcing, which requires much stronger hypotheses when singularizing successors of regular cardinals, and collapses cardinals above the cardinal whose cofinality is changed when singularizing successors of singular cardinals.

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Additional Information

Dominik Adolf
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08901

Arthur W. Apter
Affiliation: Department of Mathematics, Baruch College of CUNY, New York, New York 10010 — and — The CUNY Graduate Center, Mathematics, 365 Fifth Avenue, New York, New York 10016

Peter Koepke
Affiliation: Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität, D-53115 Bonn, Germany

Keywords: Namba forcing, singularization, Prikry forcing, measurable cardinal, Woodin cardinal, strongly compact cardinal, supercompact cardinal, consistency strength, core model
Received by editor(s): April 29, 2016
Published electronically: November 6, 2017
Additional Notes: The second author’s participation would not have been possible without the generous support of Dean Jeffrey Peck of Baruch College’s Weissman School of Arts and Sciences and Professor Warren Gordon, Chair of the Baruch College Mathematics Department, both of whom the author thanks. The second author’s research was partially supported by PSC-CUNY grants.
This paper was written in part while the authors were Visiting Fellows at the Isaac Newton Institute for Mathematical Sciences in the programme “Mathematical, Foundational and Computational Aspects of the Higher Infinite (HIF)”, held from August 19, 2015 until December 18, 2015 and funded by EPSRC grant EP/K032208/1
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2017 American Mathematical Society