Cardinals of the $P_κ(λ)$-Filter Games
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Benhamou, Tom Gitman, Victoria |
| author_facet | Benhamou, Tom Gitman, Victoria |
| contents | 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00119 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cardinals of the $P_κ(λ)$-Filter Games Benhamou, Tom Gitman, Victoria Logic 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}. |
| title | Cardinals of the $P_κ(λ)$-Filter Games |
| topic | Logic |
| url | https://arxiv.org/abs/2504.00119 |