10 found
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  1.  5
    The Real Core Model and its Scales.Daniel W. Cunningham - 1995 - Annals of Pure and Applied Logic 72 (3):213-289.
    This paper introduces the real core model K() and determines the extent of scales in this inner model. K() is an analog of Dodd-Jensen's core model K and contains L(), the smallest inner model of ZF containing the reals R. We define iterable real premice and show that Σ1∩() has the scale property when vR AD. We then prove the following Main Theorem: ZF + AD + V = K() DC. Thus, we obtain the Corollary: If ZF + AD +()L() (...)
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  2.  24
    Is There a Set of Reals Not in K?Daniel W. Cunningham - 1998 - Annals of Pure and Applied Logic 92 (2):161-210.
    We show, using the fine structure of K, that the theory ZF + AD + X R[X K] implies the existence of an inner model of ZF + AD + DC containing a measurable cardinal above its Θ, the supremum of the ordinals which are the surjective image of R. As a corollary, we show that HODK = K for some P K where K is the Dodd-Jensen Core Model relative to P. In conclusion, we show that the theory ZF (...)
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  3.  15
    Scales of Minimal Complexity in {K (\ Mathbb {R})}.Daniel W. Cunningham - 2012 - Archive for Mathematical Logic 51 (3-4):319-351.
    Using a Levy hierarchy and a fine structure theory for ${K(\mathbb{R})}$ , we obtain scales of minimal complexity in this inner model. Each such scale is obtained assuming the determinacy of only those sets of reals whose complexity is strictly below that of the scale constructed.
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  4.  9
    The Fine Structure of Real Mice.Daniel W. Cunningham - 1998 - Journal of Symbolic Logic 63 (3):937-994.
    Before one can construct scales of minimal complexity in the Real Core Model, K(R), one needs to develop the fine-structure theory of K(R). In this paper, the fine structure theory of mice, first introduced by Dodd and Jensen, is generalized to that of real mice. A relative criterion for mouse iterability is presented together with two theorems concerning the definability of this criterion. The proof of the first theorem requires only fine structure; whereas, the second theorem applies to real mice (...)
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  5.  5
    A Covering Lemma for L(ℝ).Daniel W. Cunningham - 2002 - Archive for Mathematical Logic 41 (1):49-54.
  6.  1
    A diamond-plus principle consistent with AD.Daniel W. Cunningham - forthcoming - Archive for Mathematical Logic:1-21.
    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 (...)
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  7.  21
    A Covering Lemma for HOD of K (ℝ).Daniel W. Cunningham - 2010 - Notre Dame Journal of Formal Logic 51 (4):427-442.
    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.
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  8.  7
    Strong Partition Cardinals and Determinacy in $${K}$$ K.Daniel W. Cunningham - 2015 - Archive for Mathematical Logic 54 (1-2):173-192.
    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}.
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  9.  18
    A Logical Introduction to Proof.Daniel W. Cunningham - 2012 - Springer.
    Propositional logic -- Predicate logic -- Proof strategies and diagrams -- Mathematical induction -- Set theory -- Functions -- Relations -- Core concepts in abstract algebra -- Core concepts in real analysis.
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  10.  10
    A Covering Lemma for $${K}$$.Daniel W. Cunningham - 2007 - Archive for Mathematical Logic 46 (3):197-221.
    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 (...)
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