We investigate properties of the ineffability and the Ramsey operator, and a common generalization of those that was introduced by the second author, with respect to higher indescribability, as introduced by the first author. This extends earlier investigations on the ineffability operator by James Baumgartner, and on the Ramsey operator by Qi Feng, by Philip Welch et al., and by the first author.

• 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. (shrink)

We show that $M \models \kappa$ is a completely Ramsey cardinal iff there is a Rowbottom M-ultrafilter on κ.

We study the effects of piece selection principles on cardinal arithmetic (Shelah style). As an application, we discuss questions of Abe and Usuba. In particular, we show that if, then (a) is not (λ, 2)‐distributive, and (b) does not hold.

Several situations are presented in which there is an ordinal γ such that ${\{ X \in [\gamma]^{\aleph_0} : X \cap \omega_1 \in S\,{\rm and}\, ot(X) \in T \}}$ is a stationary subset of ${[\gamma]^{\aleph_0}}$ for all stationary ${S, T\subseteq \omega_1}$ . A natural strengthening of the existence of an ordinal γ for which the above conclusion holds lies, in terms of consistency strength, between the existence of the sharp of ${H_{\omega_2}}$ and the existence of sharps for all reals. Also, an (...) optimal model separating Bounded Semiproper Forcing Axiom (BSPFA) and Bounded Martin’s Maximum (BMM) is produced and it is shown that a strong form of BMM involving only parameters from ${H_{\omega_2}}$ implies that every function from ω 1 into ω 1 is bounded on a club by a canonical function. (shrink)

We show that many large cardinal notions up to measurability can be characterized through the existence of certain filters for small models of set theory. This correspondence will allow us to obtain a canonical way in which to assign ideals to many large cardinal notions. This assignment coincides with classical large cardinal ideals whenever such ideals had been defined before. Moreover, in many important cases, relations between these ideals reflect the ordering of the corresponding large cardinal properties both under direct (...) implication and consistency strength. (shrink)