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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
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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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Asaf Karagila
Affiliation: School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, United Kingdom

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