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”
Keywords Almost Ramsey cardinal  Erdös cardinal  Indiscernibles  Core model  Supercompact Radin forcing  Radin sequence of measures  Symmetric inner model
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DOI 10.1007/s00153-008-0107-1
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References found in this work BETA

The Core Model.A. Dodd & R. Jensen - 1981 - Annals of Mathematical Logic 20 (1):43-75.
Some Results on Consecutive Large Cardinals.Arthur W. Apter - 1983 - Annals of Pure and Applied Logic 25 (1):1-17.
The Core Model for Almost Linear Iterations.Ralf-Dieter Schindler - 2002 - Annals of Pure and Applied Logic 116 (1-3):205-272.

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