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Game ideals

Annals of Pure and Applied Logic 158 (1-2):23-39 (2009)

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  1. Weak reflection principle, saturation of the nonstationary ideal on ω 1 and diamonds.Víctor Torres-pérez - 2017 - Journal of Symbolic Logic 82 (2):724-736.
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  • The Magidor function and diamond.Pierre Matet - 2011 - Journal of Symbolic Logic 76 (2):405 - 417.
    Let κ be a regular uncountable cardinal and λ be a cardinal greater than κ. We show that if 2 <κ ≤ M(κ, λ), then ◇ κ,λ holds, where M(κ, λ) equals $\lambda ^{\aleph }0$ if cf(λ) ≥ κ, and $(\lambda ^{+})^{\aleph _{0}}$ otherwise.
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  • Two-cardinal diamond star.Pierre Matet - 2014 - Mathematical Logic Quarterly 60 (4-5):246-265.
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  • Non-saturation of the nonstationary ideal on P κ (λ) in case κ≤ cf (λ)< λ.Pierre Matet - 2012 - Archive for Mathematical Logic 51 (3-4):425-432.
    Given a regular cardinal κ > ω 1 and a cardinal λ with κ ≤ cf (λ) < λ, we show that NS κ,λ | T is not λ+-saturated, where T is the set of all ${a\in P_\kappa (\lambda)}$ such that ${| a | = | a \cap \kappa|}$ and ${{\rm cf} \big( {\rm sup} (a\cap\kappa)\big) = {\rm cf} \big({\rm sup} (a)\big) = \omega}$.
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  • Non‐saturation of the non‐stationary ideal on Pκ (λ) with λ of countable cofinality.Pierre Matet - 2012 - Mathematical Logic Quarterly 58 (1-2):38-45.
    Given a regular uncountable cardinal κ and a cardinal λ > κ of cofinality ω, we show that the restriction of the non-stationary ideal on Pκ to the set of all a with equation image is not λ++-saturated . We actually prove the stronger result that there is equation image with |Q| = λ++ such that A∩B is a non-cofinal subset of Pκ for any two distinct members A, B of Q, where NGκ, λ denotes the game ideal on Pκ. (...)
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  • Ideals on $${P_{\kappa}}$$ P κ associated with games of uncountable length.Pierre Matet - 2015 - Archive for Mathematical Logic 54 (3-4):291-328.
    We study normal ideals on Pκ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${P_{\kappa} }$$\end{document} that are defined in terms of games of uncountable length.
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  • Guessing more sets.Pierre Matet - 2015 - Annals of Pure and Applied Logic 166 (10):953-990.
  • Applications of Pcf Theory to the Study of Ideals On.Pierre Matet - 2022 - Journal of Symbolic Logic 87 (3):967-994.
    Let $\kappa $ be a regular uncountable cardinal, and a cardinal greater than or equal to $\kappa $. Revisiting a celebrated result of Shelah, we show that if is close to $\kappa $ and (= the least size of a cofinal subset of ) is greater than, then can be represented (in the sense of pcf theory) as a pseudopower. This can be used to obtain optimal results concerning the splitting problem. For example we show that if and, then no (...)
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