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All uncountable cardinals in the Gitik model are almost Ramsey and carry Rowbottom filters
Author(s) -
Apter Arthur W.,
Dimitriou Ioanna M.,
Koepke Peter
Publication year - 2016
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201400050
Subject(s) - uncountable set , mathematics , regular cardinal , class (philosophy) , limit (mathematics) , cardinal number (linguistics) , consistency (knowledge bases) , discrete mathematics , combinatorics , computer science , mathematical analysis , artificial intelligence , philosophy , linguistics , countable set
Using the analysis developed in our earlier paper [5][A. W. Apter, 2014], we show that every uncountable cardinal in Gitik's model of [8][M. Gitik, 1980] in which all uncountable cardinals are singular is almost Ramsey and is also a Rowbottom cardinal carrying a Rowbottom filter. We assume that the model of [8][M. Gitik, 1980] is constructed from a proper class of strongly compact cardinals, each of which is a limit of measurable cardinals. Our work consequently reduces the best previously known upper bound in consistency strength for the theory ZF + “All uncountable cardinals are singular” + “Every uncountable cardinal is both almost Ramsey and a Rowbottom cardinal carrying a Rowbottom filter”.