All uncountable cardinals in the Gitik model are almost Ramsey and carry Rowbottom filters | Zendy

Apter Arthur W. | Zendy; Dimitriou Ioanna M. | Zendy; Koepke Peter | Zendy

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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”.

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