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

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• 作者：I. Sharpe^a ; P.D. Welch ; ^a ; ^{p.welch@bristol.ac.uk"" rel=""nofollow}
• 关键词：Ramsey cardinal ; Chang property ; Erdő ; s cardinal ; Core model
• 刊名：Annals of Pure and Applied Logic
• 出版年：2011
• 出版时间：November 2011
• 年：2011
• 卷：162
• 期：11
• 页码：863-902
• 全文大小：592 K

文摘

• 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:

Theorem

If 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.