# A covering lemma for $${K}$$

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

 Abstract The Dodd–Jensen Covering Lemma states that “if there is no inner model with a measurable cardinal, then for any uncountable set of ordinals X, there is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Y\in K}$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X\subseteq Y}$$\end{document} and |X| = |Y|”. Assuming ZF+AD alone, we establish the following analog: If there is no inner model with an \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} –complete measurable cardinal, then the real core model \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} is a “very good approximation” to the universe of sets V; that is, \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} and V have exactly the same sets of reals and for any set of ordinals X with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${|{X}|\ge\Theta}$$\end{document}, there is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Y\in K}$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X\subseteq Y}$$\end{document} and |X| = |Y|. Here \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} is the set of reals and \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 \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}. Keywords Mathematics   Algebra   Mathematics, general   Mathematical Logic and Foundations Categories (categorize this paper) ISBN(s) DOI 10.1007/s00153-007-0040-8 Options Edit this record Mark as duplicate Export citation Request removal from index

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## References found in this work BETA

The Higher Infinite.Akihiro Kanamori - 2000 - Studia Logica 65 (3):443-446.
The Core Model.A. Dodd & R. Jensen - 1981 - Annals of Mathematical Logic 20 (1):43-75.
The Real Core Model and its Scales.Daniel W. Cunningham - 1995 - Annals of Pure and Applied Logic 72 (3):213-289.
Is There a Set of Reals Not in K?Daniel W. Cunningham - 1998 - Annals of Pure and Applied Logic 92 (2):161-210.
A Covering Lemma for L(ℝ).Daniel W. Cunningham - 2002 - Archive for Mathematical Logic 41 (1):49-54.

## Citations of this work BETA

Strong Partition Cardinals and Determinacy in $${K}$$ K.Daniel W. Cunningham - 2015 - Archive for Mathematical Logic 54 (1-2):173-192.