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Proceedings of the American Mathematical Society

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The Morris model

Author: Asaf Karagila
Journal: Proc. Amer. Math. Soc. 148 (2020), 1311-1323
MSC (2010): Primary 03E25; Secondary 03E35
Published electronically: September 20, 2019
MathSciNet review: 4055957
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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 $\kappa$”).

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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

Keywords: Axiom of choice, symmetric extensions, iterations of symmetric extensions, countable union theorem
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
Article copyright: © Copyright 2019 American Mathematical Society