Cardinals of the $P_κ(λ)$-Filter Games

TL;DR

This work develops a two-cardinal generalization of filter-extension games to filters on $P_\kappa(\lambda)$ and uses winning strategies for the judge to characterize a broad spectrum of large-cardinal notions. It extends the one-cardinal framework of Holy–Schlicht and Nielsen–Welch, connects finite-length and infinite-length games to two-cardinal indescribability and ineffability, and relates game strategies to generic versions of supercompactness and strong compactness through weak amenability and forcing. The authors establish equi- or implication-relationships between game outcomes and large-cardinal properties, including $\lambda$-strong compactness, $\lambda$-supercompactness, and completely $\lambda$-ineffable cardinals, and they construct precipitous ideals on $P_\kappa(\lambda)$ from winning strategies. They also extend these ideas to the measurable case via trees of $M$-ultrafilters, refine previous results of Foreman–Magidor–Zeman, and address open problems in the two-cardinal setting, broadening the landscape of filter-game techniques in large-cardinal theory.

Abstract

We investigate forms of filter extension properties in the two-cardinal setting involving filters on $P_κ(λ)$. We generalize the filter games introduced by Holy and Schlicht in \cite{HolySchlicht:HierarchyRamseyLikeCardinals} to filters on $P_κ(λ)$ and show that the existence of a winning strategy for Player II in a game of a certain length can be used to characterize several large cardinal notions such as: $λ$-super/strongly compact cardinals, $λ$-completely ineffable cardinals, nearly $λ$-super/strongly compact cardinals, and various notions of generic super and strong compactness. We generalize a result of Nielson from \cite{NielsenWelch:games_and_Ramsey-like_cardinals} connecting the existence of a winning strategy for Player II in a game of finite length and two-cardinal indescribability. We generalize the result of \cite{ForMagZem} to construct a fine $κ$-complete precipitous ideal on $P_κ(λ)$ from a winning strategy for Player II in a game of length $ω$. Finally, we improve Theorems 1.2 and 1.4 from \cite{ForMagZem} and partially answer questions Q.1 and Q.2 from \cite{ForMagZem}.

Figures (1)

• Figure 1: Implication table

Theorems & Definitions (128)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem
• Theorem
• Theorem
• Theorem
• Theorem
• Theorem 1.5