Chang’s conjecture and semiproperness of nonreasonable posets
Chang’s conjecture and semiproperness of nonreasonable posets
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张猜想与非合理偏序集的半正确性
DOI：
10.1007/s00605-018-1182-y
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Sean D. Cox
中科院分区：
文献类型：
--
作者：
Sean D. Cox
令 Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document} 表示添加科恩实数的偏序集然后通过 ([ω2]ω)V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} 的补集射出俱乐部\begin{document}$$\big ( [\omega _2]^\omega \big )^V$$\end{document} 具有可数条件。我们证明来自 Todorčević 和 Torres-Pérez (MLQ Math Log Q 58(4–5):342–347, 2012) 的 Strong Chang 猜想的版本意味着 Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} 的半正确性\usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document} ，以及 Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} 的半正确性\usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document} - 事实上，任何在 Foreman 和 Magidor 意义上足够不合理的偏序集的半正确性（Ann Pure Appl Log） 76(1):47–97, 1995）——暗示来自 Woodin 的强 Chang 猜想（确定性公理、强制公理和非平稳理想，de Gruyter Series in Logic and its applications, 2nd ed., vol 1, Walter de Gruyter GmbH & Co. KG, Berlin, 2010）和 Todorčević（猜想） Rado 和 Chang 的《基数算术》，Kluwer Acadamic，多德雷赫特，1993 年）。特别是 Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb 的半正确性{Q}$$\end{document} 具有很大的基数强度，这回答了 Friedman 和 Krueger 的问题 (Trans. Am. Math. Soc. 359(5):2407–2420, 2007)。我们工作的一个推论是 Todorčević 和 Torres-Pérez 的 Strong Chang 猜想 (MLQ Math Log Q 58(4–5):342–347, 2012) 的版本并不意味着 ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} 上存在陡峭理想\usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{文档}$$\omega _1$$\end{文档}。
Let Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document} denote the poset which adds a Cohen real then shoots a club through the complement of ([ω2]ω)V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ( [\omega _2]^\omega \big )^V$$\end{document} with countable conditions. We prove that the version of Strong Chang’s conjecture from Todorčević and Torres-Pérez (MLQ Math Log Q 58(4–5):342–347, 2012) implies semiproperness of Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document}, and that semiproperness of Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document}—in fact semiproperness of any poset which is sufficiently nonreasonable in the sense of Foreman and Magidor (Ann Pure Appl Log 76(1):47–97, 1995)—implies the version of strong Chang’s conjecture from Woodin (The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter Series in Logic and its Applications, 2nd ed., vol 1, Walter de Gruyter GmbH & Co. KG, Berlin, 2010) and Todorčević (Conjectures of Rado and Chang and cardinal arithmetic, Kluwer Acadamic, Dordrecht, 1993). In particular, semiproperness of Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document} has large cardinal strength, which answers a question of Friedman and Krueger (Trans. Am. Math. Soc. 359(5):2407–2420, 2007). One corollary of our work is that the version of Strong Chang’s Conjecture from Todorčević and Torres-Pérez (MLQ Math Log Q 58(4–5):342–347, 2012) does not imply the existence of a precipitous ideal on ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _1$$\end{document}.
$omega_2$的反射原理和半平稳反射原理
影响因子：
0.7
作者：
S. Haerting;A. Marciniak-Czochra and I. Takagi;Toshimichi Usuba
通讯作者：
Toshimichi Usuba