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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Morris model
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by Asaf Karagila PDF
Proc. Amer. Math. Soc. 148 (2020), 1311-1323 Request permission

Abstract:

Douglass B. Morris announced in 1970 that it is consistent with $\mathsf {ZF}$ that “For every $\alpha$, there exists a set $A_\alpha$ which is the countable union of countable sets, and $\mathcal P(A_\alpha )$ can be partitioned into $\aleph _\alpha$ non-empty sets”. The result was never published in a journal (it was proved in full in Morris’ dissertation) and seems to have been lost, save a mention in Jech’s “Axiom of Choice”. We provide a proof using modern tools derived from recent work of the author. We also prove a new preservation theorem for general products of symmetric systems, which we use to obtain the consistency of Dependent Choice with the above statement (replacing “countable union of countable sets” by “union of $\kappa$ sets of size $\kappa$”).
References
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Additional Information
  • Asaf Karagila
  • Affiliation: School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, United Kingdom
  • MR Author ID: 1075269
  • ORCID: 0000-0003-1289-0904
  • Email: karagila@math.huji.ac.il
  • Received by editor(s): November 27, 2018
  • Received by editor(s) in revised form: February 13, 2019, and July 22, 2019
  • Published electronically: September 20, 2019
  • Additional Notes: The author was supported by the Royal Society grant no. NF170989
  • Communicated by: Heike Mildenberger
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 1311-1323
  • MSC (2010): Primary 03E25; Secondary 03E35
  • DOI: https://doi.org/10.1090/proc/14770
  • MathSciNet review: 4055957