Greatly Erdős 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 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. 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 ω 3 = ( ω 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 + ∧ ¬ □ ω 2 then there is an inner model with a strong cardinal. 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.