June 2011 Ramsey-like cardinals II
Victoria Gitman, P. D Welch
J. Symbolic Logic 76(2): 541-560 (June 2011). DOI: 10.2178/jsl/1305810763


This paper continues the study of the Ramsey-like large cardinals introduced in [5] and [14]. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only α-many iterations for some countable ordinal α. Here we study such α-iterable cardinals. We show that the α-iterable cardinals form a strict hierarchy for α≤ω₁, that they are downward absolute to L for α <ω₁L, and that the consistency strength of Schindler's remarkable cardinals is strictly between 1-iterable and 2-iterable cardinals.

We show that the strongly Ramsey and super Ramsey cardinals from [5] are downward absolute to the core model K. Finally, we use a forcing argument from a strongly Ramsey cardinal to separate the notions of Ramsey and virtually Ramsey cardinals. These were introduced in [14] as an upper bound on the consistency strength of the Intermediate Chang's Conjecture.


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Victoria Gitman. P. D Welch. "Ramsey-like cardinals II." J. Symbolic Logic 76 (2) 541 - 560, June 2011. https://doi.org/10.2178/jsl/1305810763


Published: June 2011
First available in Project Euclid: 19 May 2011

zbMATH: 1222.03055
MathSciNet: MR2830435
Digital Object Identifier: 10.2178/jsl/1305810763

Rights: Copyright © 2011 Association for Symbolic Logic


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Vol.76 • No. 2 • June 2011
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