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

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 alpha-weakly Eras 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 Erdos 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 omega(3) = (omega(+)(2))(K) 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 CC(+) boolean AND inverted left perpendicular square(omega 2), then there is an inner model with a strong cardinal.

We define an alpha-Jonsson hierarchy to parallel the alpha-Ramsey hierarchy, and show that kappa being alpha-Jonsson implies that it is alpha-Ramsey in the core model. (C) 2011 Elsevier B.V. All rights reserved.