Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties

Annals of Pure and Applied Logic 162 (11):863-902 (2011)
Ian Sharpe
Charles Sturt University
• We define a notion of order of indiscernibility type of a structure by analogy with Mitchell order on measures; we use this to define a hierarchy of strong axioms of infinity defined through normal filters, the α-weakly Erdős hierarchy. The filters in this hierarchy can be seen to be generated by sets of ordinals where these indiscernibility orders on structures dominate the canonical functions.• The limit axiom of this is that of greatly Erdős and we use it to calibrate some strengthenings of the Chang property, one of which, CC+, is equiconsistent with a Ramsey cardinal, and implies that where K is the core model built with non-overlapping extenders — if it is rigid, and others which are a little weaker. As one corollary we have:TheoremIf then there is an inner model with a strong cardinal. • We define an α-Jónsson hierarchy to parallel the α-Ramsey hierarchy, and show that κ being α-Jónsson implies that it is α-Ramsey in the core model
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DOI 10.1016/j.apal.2011.04.002
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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.
Ramsey-Like Cardinals.Victoria Gitman - 2011 - Journal of Symbolic Logic 76 (2):519 - 540.
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On the Size of Closed Unbounded Sets.James E. Baumgartner - 1991 - Annals of Pure and Applied Logic 54 (3):195-227.
Adding Closed Unbounded Subsets of Ω₂ with Finite Forcing.William J. Mitchell - 2005 - Notre Dame Journal of Formal Logic 46 (3):357-371.

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Citations of this work BETA

In Memoriam: James Earl Baumgartner.J. A. Larson - 2017 - Archive for Mathematical Logic 56 (7-8):877-909.
Games and Ramsey-Like Cardinals.Dan Saattrup Nielsen & Philip Welch - 2019 - Journal of Symbolic Logic 84 (1):408-437.

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