Archive for Mathematical Logic 47 (7-8):769-783 (2008)
Abstract |
We examine combinatorial aspects and consistency strength properties of almost Ramsey cardinals. Without the Axiom of Choice, successor cardinals may be almost Ramsey. From fairly mild supercompactness assumptions, we construct a model of ZF + ${\neg {\rm AC}_\omega}$ in which every infinite cardinal is almost Ramsey. Core model arguments show that strong assumptions are necessary. Without successors of singular cardinals, we can weaken this to an equiconsistency of the following theories: “ZFC + There is a proper class of regular almost Ramsey cardinals”, and “ZF + DC + All infinite cardinals except possibly successors of singular cardinals are almost Ramsey”
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Keywords | Almost Ramsey cardinal Erdös cardinal Indiscernibles Core model Supercompact Radin forcing Radin sequence of measures Symmetric inner model |
Categories | (categorize this paper) |
DOI | 10.1007/s00153-008-0107-1 |
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References found in this work BETA
Some Results on Consecutive Large Cardinals.Arthur W. Apter - 1983 - Annals of Pure and Applied Logic 25 (1):1-17.
Some New Upper Bounds in Consistency Strength for Certain Choiceless Large Cardinal Patterns.Arthur W. Apter - 1992 - Archive for Mathematical Logic 31 (3):201-205.
The Core Model for Almost Linear Iterations.Ralf-Dieter Schindler - 2002 - Annals of Pure and Applied Logic 116 (1-3):205-272.
The Strength of Choiceless Patterns of Singular and Weakly Compact Cardinals.Daniel Busche & Ralf Schindler - 2009 - Annals of Pure and Applied Logic 159 (1-2):198-248.
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Citations of this work BETA
All Uncountable Cardinals in the Gitik Model Are Almost Ramsey and Carry Rowbottom Filters.Arthur W. Apter, Ioanna M. Dimitriou & Peter Koepke - 2016 - Mathematical Logic Quarterly 62 (3):225-231.
How Many Normal Measures Can ℵmath Image Carry?Arthur W. Apter - 2010 - Mathematical Logic Quarterly 56 (2):164-170.
Remarks on Gitik's Model and Symmetric Extensions on Products of the Lévy Collapse.Amitayu Banerjee - 2020 - Mathematical Logic Quarterly 66 (3):259-279.
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