Two-cardinal ideal operators and indescribability

TL;DR

This paper explores the relationships between two-cardinal Ramseyness, ineffability, and indescribability, generalizing prior results and establishing hierarchies in the context of supercompactness ultrafilters.

Contribution

It introduces new notions of two-cardinal Ramseyness and ineffability, and studies their hierarchies and connections to transfinite two-cardinal indescribability.

Findings

Established hierarchies for two-cardinal Ramseyness and ineffability.

Connected these hierarchies to transfinite two-cardinal indescribability.

Generalized previous results by Baumgartner, Feng, and others.

Abstract

A well-known version of Rowbottom's theorem for supercompactness ultrafilters leads naturally to notions of two-cardinal Ramseyness and corresponding normal ideals introduced herein. Generalizing results of Baumgartner [7, 8], Feng [22] and the first author [16, 17], we study the hierarchies associated with a particular version of two-cardinal Ramseyness and a strong version of two-cardinal ineffability, as well as the relationships between these hierarchies and a natural notion of transfinite two-cardinal indescribability.