Projects per year
Abstract
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 nonempty 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 language  English 

Pages (fromto)  13111323 
Number of pages  13 
Journal  Proceedings of the American Mathematical Society 
Volume  148 
Issue number  3 
Early online date  20 Sep 2019 
DOIs  
Publication status  Published  Mar 2020 
Keywords
 Axiom of choice
 symmetric extensions
 iterations of symmetric extensions
 countable union theorem
Profiles

Asaf Karagila
 School of Mathematics  UKRI Future Leaders Fellow
 Logic  Member
Person: Research Group Member, Academic, Teaching & Research
Projects
 1 Finished

High Forcing Axioms: Forcing Axioms for the Uncountable. Newton International Fellowship
Aspero, D. & Karagila, A.
1/03/18 → 31/03/20
Project: Fellowship