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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We wish to thank Ralf Schindler for his insightful comments concerning Lemma 4 and Theorem 2.
A. W. Apter’s research was partially supported by PSC-CUNY grants and CUNY Collaborative Incentive grants. In addition, he wishes to thank the members of the mathematical logic group in Bonn for all of the hospitality shown him during his spring 2007 sabbatical visit to the Mathematisches Institut.
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Apter, A.W., Koepke, P. Making all cardinals almost Ramsey. Arch. Math. Logic 47, 769–783 (2008). https://doi.org/10.1007/s00153-008-0107-1
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DOI: https://doi.org/10.1007/s00153-008-0107-1
Keywords
- Almost Ramsey cardinal
- Erdös cardinal
- Indiscernibles
- Core model
- Supercompact Radin forcing
- Radin sequence of measures
- Symmetric inner model