The Morris model

Asaf Karagila

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Douglass B. Morris announced in 1970 that it is consistent with ZF that "For every α, there exists a set Aα which is the countable union of countable sets, and P(Aα) can be partitioned into ℵα non-empty sets". The result was never published in a journal, 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 κ sets of size κ").
Original languageEnglish
Pages (from-to)1311-1323
Number of pages13
JournalProceedings of the American Mathematical Society
Issue number3
Early online date20 Sep 2019
Publication statusPublished - Mar 2020


  • Axiom of choice
  • symmetric extensions
  • iterations of symmetric extensions
  • countable union theorem

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