Ramsey-like cardinals

This paper continues the study of the Ramsey-like large cardinals introduced in [Git09] and [WS08]. Ramsey-like cardinals are defined by generalizing the “existence of elementary embeddings” characterization of Ramsey cardinals. A cardinal κ is Ramsey if and only if every subset of κ can be put into a κ-size transitive model of ZFC for which there exists a weakly amenable countably complete ultrafilter. Such ultrafilters 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 and they are downward absolute to L for α < ωL 1 . We show that the strongly Ramsey and super Ramsey cardinals from [Git09] are downward absolute to the core model K. Finally, we use a forcing argument from a strongly Ramsey cardinal to separate the notions o...

• Type: Journal
• Year: 2011
• Where: JSYML
• Authors: Victoria Gitman
• Keywords: Consistency Strength, Core Model, JSYML 2011, Strict Hierarchy