Two-cardinal ideal operators and indescribability
Brent Cody, Philip White
TL;DR
The paper formulates a comprehensive two-cardinal framework for Ramseyness and ineffability via ideal operators ${\mathcal{I}}$ and ${\mathcal{R}}$, and pairs them with a two-cardinal notion of indescribability given by $\Pi^1_\xi(\kappa,A)$. It develops transfinite hierarchies by iterating these operators, analyzes their generation via sub-ideals, and establishes hierarchy results using canonical functions. A key contribution is showing that many ${\mathcal{O}}^\alpha(\Pi^1_\xi(\kappa,A))$ can be obtained from two smaller sub-ideals, extending Baumgartner–Feng-style results to the two-cardinal setting, and linking these to the strength of two-cardinal large-cardinal properties and the existence of indiscernibles. The work provides a systematic toolkit for comparing the consistency strength of two-cardinal ineffability/Ramseyness notions and their indescribability, with further developments (e.g., CLHZ2024) confirming analogous single-cardinal results. Overall, the paper deepens the structural understanding of two-cardinal large-cardinal hierarchies and their descriptive-set-theoretic characterizations.
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.