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A covering lemma for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${K(\mathbb {R})}$$\end{document} [Book Review]

Daniel W. Cunningham

Archive for Mathematical Logic 46 (3-4):197-221 (2007)

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Strong partition cardinals and determinacy in $${K}$$ K.Daniel W. Cunningham - 2015 - Archive for Mathematical Logic 54 (1-2):173-192.details We prove within K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${K}$$\end{document} that the axiom of determinacy is equivalent to the assertion that for each ordinal λ λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa > \lambda}$$\end{document}. Here Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta}$$\end{document} is the supremum of the ordinals which are the surjective image of the set of reals R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document}. Download Export citation Bookmark
A Covering Lemma for HOD of K (ℝ).Daniel W. Cunningham - 2010 - Notre Dame Journal of Formal Logic 51 (4):427-442.details Working in ZF+AD alone, we prove that every set of ordinals with cardinality at least Θ can be covered by a set of ordinals in HOD of K (ℝ) of the same cardinality, when there is no inner model with an ℝ-complete measurable cardinal. Here ℝ is the set of reals and Θ is the supremum of the ordinals which are the surjective image of ℝ. Download Export citation Bookmark
A Diamond Principle Consistent with AD.Daniel Cunningham - 2017 - Notre Dame Journal of Formal Logic 58 (3):397-407.details We present a diamond principle ◊R concerning all subsets of Θ, the supremum of the ordinals that are the surjective image of R. We prove that ◊R holds in Steel’s core model K, a canonical inner model for determinacy. Download Export citation Bookmark 1 citation
A diamond-plus principle consistent with AD.Daniel W. Cunningham - 2020 - Archive for Mathematical Logic 59 (5-6):755-775.details After showing that \ refutes \ for all regular cardinals \, we present a diamond-plus principle \ concerning all subsets of \. Using a forcing argument, we prove that \ holds in Steel’s core model \}}\), an inner model in which the axiom of determinacy can hold. The combinatorial principle \ is then extended, in \}}\), to successor cardinals \ and to certain cardinals \ that are not ineffable. Here \ is the supremum of the ordinals that are the surjective (...) image of the set of reals \. (shrink) Download Export citation Bookmark

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