45 found
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James Cummings [45]James W. Cummings [3]
  1.  35
    Squares, Scales and Stationary Reflection.James Cummings, Matthew Foreman & Menachem Magidor - 2001 - Journal of Mathematical Logic 1 (01):35-98.
    Since the work of Gödel and Cohen, which showed that Hilbert's First Problem was independent of the usual assumptions of mathematics, there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond and square discovered by Jensen. Simultaneously, attempts have been made to find suitable (...)
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  2.  4
    Scales, Squares and Reflection.James Cummings, Matthew Foreman & Menachem Magidor - 2001 - Journal of Mathematical Logic 1 (1):35-98.
    Since the work of Gödel and Cohen, which showed that Hilbert's First Problem was independent of the usual assumptions of mathematics, there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond and square discovered by Jensen. Simultaneously, attempts have been made to find suitable (...)
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  3.  26
    Canonical Structure in the Universe of Set Theory: Part Two.James Cummings, Matthew Foreman & Menachem Magidor - 2006 - Annals of Pure and Applied Logic 142 (1):55-75.
    We prove a number of consistency results complementary to the ZFC results from our paper [J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of set theory: part one, Annals of Pure and Applied Logic 129 211–243]. We produce examples of non-tightly stationary mutually stationary sequences, sequences of cardinals on which every sequence of sets is mutually stationary, and mutually stationary sequences not concentrating on a fixed cofinality. We also give an alternative proof for the consistency of the (...)
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  4.  24
    Canonical Structure in the Universe of Set Theory: Part One.James Cummings, Matthew Foreman & Menachem Magidor - 2004 - Annals of Pure and Applied Logic 129 (1-3):211-243.
    We start by studying the relationship between two invariants isolated by Shelah, the sets of good and approachable points. As part of our study of these invariants, we prove a form of “singular cardinal compactness” for Jensen's square principle. We then study the relationship between internally approachable and tight structures, which parallels to a certain extent the relationship between good and approachable points. In particular we characterise the tight structures in terms of PCF theory and use our characterisation to prove (...)
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  5.  22
    Identity Crises and Strong Compactness : II. Strong Cardinals.Arthur W. Apter & James Cummings - 2001 - Archive for Mathematical Logic 40 (1):25-38.
    . From a proper class of supercompact cardinals, we force and obtain a model in which the proper classes of strongly compact and strong cardinals precisely coincide. In this model, it is the case that no strongly compact cardinal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\kappa$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $2^\kappa = \kappa^+$\end{document} supercompact.
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  6.  7
    Notes on Singular Cardinal Combinatorics.James Cummings - 2005 - Notre Dame Journal of Formal Logic 46 (3):251-282.
    We present a survey of combinatorial set theory relevant to the study of singular cardinals and their successors. The topics covered include diamonds, squares, club guessing, forcing axioms, and PCF theory.
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  7.  16
    Diagonal Prikry Extensions.James Cummings & Matthew Foreman - 2010 - Journal of Symbolic Logic 75 (4):1383-1402.
  8.  28
    Cardinal Invariants Above the Continuum.James Cummings & Saharon Shelah - 1995 - Annals of Pure and Applied Logic 75 (3):251-268.
    We prove some consistency results about and δ, which are natural generalisations of the cardinal invariants of the continuum and . We also define invariants cl and δcl, and prove that almost always = cl and = cl.
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  9.  8
    Collapsing the Cardinals of HOD.James Cummings, Sy David Friedman & Mohammad Golshani - 2015 - Journal of Mathematical Logic 15 (2):1550007.
    Assuming that GCH holds and [Formula: see text] is [Formula: see text]-supercompact, we construct a generic extension [Formula: see text] of [Formula: see text] in which [Formula: see text] remains strongly inaccessible and [Formula: see text] for every infinite cardinal [Formula: see text]. In particular the rank-initial segment [Formula: see text] is a model of ZFC in which [Formula: see text] for every infinite cardinal [Formula: see text].
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  10.  22
    Identity Crises and Strong Compactness.Arthur W. Apter & James Cummings - 2000 - Journal of Symbolic Logic 65 (4):1895-1910.
    Combining techniques of the first author and Shelah with ideas of Magidor, we show how to get a model in which, for fixed but arbitrary finite n, the first n strongly compact cardinals κ 1 ,..., κ n are so that κ i for i = 1,..., n is both the i th measurable cardinal and κ + i supercompact. This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah.
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  11.  11
    The Eightfold Way.James Cummings, Sy-David Friedman, Menachem Magidor, Assaf Rinot & Dima Sinapova - 2018 - Journal of Symbolic Logic 83 (1):349-371.
    Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at${\kappa ^{ + + }}$, assuming that$\kappa = {\kappa ^{ < \kappa }}$and there is a weakly compact cardinal aboveκ.If in additionκis supercompact then we can forceκto be${\aleph _\omega }$in the extension. The proofs combine the techniques of adding and then destroying (...)
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  12.  9
    Aronszajn and Kurepa Trees.James Cummings - 2018 - Archive for Mathematical Logic 57 (1-2):83-90.
    Monroe Eskew and \, 2016. https://mathoverflow.net/questions/217951/tree-properties-on-omega-1-and-omega-2) asked whether the tree property at \ implies there is no Kurepa tree. We prove that the tree property at \ is consistent with the existence of \-trees with as many branches as desired.
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  13.  27
    A Global Version of a Theorem of Ben-David and Magidor.Arthur W. Apter & James Cummings - 2000 - Annals of Pure and Applied Logic 102 (3):199-222.
    We prove a consistency result about square principles and stationary reflection which generalises the result of Ben-David and Magidor [4].
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  14.  21
    The Non-Compactness of Square.James Cummings, Matthew Foreman & Menachem Magidor - 2003 - Journal of Symbolic Logic 68 (2):637-643.
  15.  13
    Possible Behaviours for the Mitchell Ordering.James Cummings - 1993 - Annals of Pure and Applied Logic 65 (2):107-123.
    We use mixture of forcing and inner models techniques to get some results on the possible behaviours of the Mitchell ordering at a measurable к.
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  16.  6
    Small Universal Families of Graphs on ℵω+ 1.James Cummings, Mirna Džamonja & Charles Morgan - 2016 - Journal of Symbolic Logic 81 (2):541-569.
  17.  34
    Some Results in Polychromatic Ramsey Theory.Uri Abraham, James Cummings & Clifford Smyth - 2007 - Journal of Symbolic Logic 72 (3):865 - 896.
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  18.  31
    More on Full Reflection Below $${\aleph_\omega}$$.James Cummings & Dorshka Wylie - 2010 - Archive for Mathematical Logic 49 (6):659-671.
    Jech and Shelah in J Symb Log, 55, 822–830 (1990) studied full reflection below ${\aleph_\omega}$ , and produced a model in which the extent of full reflection is maximal in a certain sense. We produce a model in which full reflection is maximised in a different direction.
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  19.  17
    Possible Behaviours for the Mitchell Ordering II.James Cummings - 1994 - Journal of Symbolic Logic 59 (4):1196-1209.
    We analyse the Mitchell ordering in a model where κ is P 2 κ-hypermeasurable and $2^{2^\kappa} > (2^\kappa)^+$.
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  20.  19
    An L-Like Model Containing Very Large Cardinals.Arthur W. Apter & James Cummings - 2008 - Archive for Mathematical Logic 47 (1):65-78.
    We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with a strong form of diamond and a version of square consistent with supercompactness. This generalises a result due to the first author. There are no restrictions in our model on the structure of the class of supercompact cardinals.
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  21.  17
    □ On the Singular Cardinals.James Cummings & Sy-David Friedman - 2008 - Journal of Symbolic Logic 73 (4):1307-1314.
    We give upper and lower bounds for the consistency strength of the failure of a combinatorial principle introduced by Jensen. "Square on singular cardinals".
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  22.  20
    Strong Ultrapowers and Long Core Models.James Cummings - 1993 - Journal of Symbolic Logic 58 (1):240-248.
  23.  65
    Blowing Up the Power Set of the Least Measurable.Arthur W. Apter & James Cummings - 2002 - Journal of Symbolic Logic 67 (3):915-923.
    We prove some results related to the problem of blowing up the power set of the least measurable cardinal. Our forcing results improve those of [1] by using the optimal hypothesis.
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  24.  6
    Normal Measures on a Tall Cardinal.Arthur W. Apter & James Cummings - 2019 - Journal of Symbolic Logic 84 (1):178-204.
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  25. REVIEWS-Ten Papers.Arthur Apter & James W. Cummings - 2000 - Bulletin of Symbolic Logic 6 (1):86-88.
  26. Foundations of Mathematics.Andrés Eduardo Caicedo, James Cummings, Peter Koellner & Paul B. Larson (eds.) - 2016 - American Mathematical Society.
    This volume contains the proceedings of the Logic at Harvard conference in honor of W. Hugh Woodin's 60th birthday, held March 27–29, 2015, at Harvard University. It presents a collection of papers related to the work of Woodin, who has been one of the leading figures in set theory since the early 1980s. The topics cover many of the areas central to Woodin's work, including large cardinals, determinacy, descriptive set theory and the continuum problem, as well as connections between set (...)
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  27.  13
    A Model in Which Every Boolean Algebra has Many Subalgebras.James Cummings & Saharon Shelah - 1995 - Journal of Symbolic Logic 60 (3):992-1004.
    We show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebra B has an irredundant subset A such that 2 |A| = 2 |B| . This implies in particular that B has 2 |B| subalgebras. We also discuss some more general problems about subalgebras and free subsets of an algebra. The result on the number of subalgebras in a Boolean algebra solves a question of Monk from [6]. The paper is intended to be accessible (...)
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  28.  9
    Alexander Razborov, Flag Algebras. Journal of Symbolic Logic, Vol. 72 , No. 4, Pp. 1239–1282.James Cummings - 2018 - Bulletin of Symbolic Logic 24 (1):107-108.
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  29.  51
    Arthur W. Apter. On the Least Strongly Compact Cardinal. Israel Journal of Mathematics, Vol. 35 , Pp. 225–233. - Arthur W. Apter. Measurability and Degrees of Strong Compactness. The Journal of Symbolic Logic, Vol. 46 , Pp. 249–254. - Arthur W. Apter. A Note on Strong Compactness and Supercompactness. Bulletin of the London Mathematical Society, Vol. 23 , Pp. 113–115. - Arthur W. Apter. On the First N Strongly Compact Cardinals. Proceedings of the American Mathematical Society, Vol. 123 , Pp. 2229–2235. - Arthur W. Apter and Saharon Shelah. On the Strong Equality Between Supercompactness and Strong Compactness.. Transactions of the American Mathematical Society, Vol. 349 , Pp. 103–128. - Arthur W. Apter and Saharon Shelah. Menas' Result is Best Possible. Ibid., Pp. 2007–2034. - Arthur W. Apter. More on the Least Strongly Compact Cardinal. Mathematical Logic Quarterly, Vol. 43 , Pp. 427–430. - Arthur W. Apter. Laver Indestructibility and the Class of Compact Cardinals. The Journal of Sy. [REVIEW]James W. Cummings - 2000 - Bulletin of Symbolic Logic 6 (1):86-89.
  30.  7
    Coherent Sequences Versus Radin Sequences.James Cummings - 1994 - Annals of Pure and Applied Logic 70 (3):223-241.
    We attempt to make a connection between the sequences of measures used to define Radin forcing and the coherent sequences of extenders which are the basis of modern inner model theory. We show that in certain circumstances we can read off sequences of measures as defined by Radin from coherent sequences of extenders, and that we can define Radin forcing directly from a coherent extender sequence and a sequence of ordinals; this generalises Mitchell's construction of Radin forcing from a coherent (...)
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  31.  15
    Diamond and Antichains.James Cummings & Ernest Schimmerling - 2005 - Archive for Mathematical Logic 44 (1):71-76.
    It is obvious that ♦ implies the existence of an antichain of stationary sets of cardinality which is the largest possible cardinality. We show that the obvious antichain is not maximal and find a less obvious extension of it by ℵ2 more stationary sets.
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  32.  13
    Gitik Moti. On the Mitchell and Rudin-Keisler Orderings of Ultrafilters. Annals of Pure and Applied Logic, Vol. 39 , Pp. 175–197. [REVIEW]James Cummings - 1995 - Journal of Symbolic Logic 60 (1):338-339.
  33.  14
    Gitik Moti. The Strength of the Failure of the Singular Cardinal Hypothesis. Annals of Pure and Applied Logic, Vol. 51 , Pp. 215–240. [REVIEW]James Cummings - 1995 - Journal of Symbolic Logic 60 (1):340-340.
  34.  11
    Itay Neeman. Aronszajn Trees and Failure of the Singular Cardinal Hypothesis. Journal of Mathematical Logic, Vol. 9, No. 1 , Pp. 139–157. - Dima Sinapova. The Tree Property at אּω+1. Journal of Symbolic Logic, Vol. 77, No. 1 , Pp. 279–290. - Dima Sinapova. The Tree Property and the Failure of SCH at Uncountable Cofinality. Archive for Mathematical Logic, Vol. 51, No. 5-6 , Pp. 553–562. - Dima Sinapova. The Tree Property and the Failure of the Singular Cardinal Hypothesis at אּω2. Journal of Symbolic Logic, Vol. 77, No. 3 , Pp. 934–946. - Spencer Unger. Aronszajn Trees and the Successors of a Singular Cardinal. Archive for Mathematical Logic, Vol. 52, No. 5-6 , Pp. 483–496. - Itay Neeman. The Tree Property Up to אּω+1. Journal of Symbolic Logic. Vol. 79, No. 2 , Pp. 429–459. [REVIEW]James Cummings - 2015 - Bulletin of Symbolic Logic 21 (2):188-192.
  35.  18
    Moti Gitik and Menachem Magidor. The Singular Cardinal Hypothesis Revisited. Set Theory of the Continuum, Edited by H. Judah, W. Just, and H. Woodin, Mathematical Sciences Research Institute Publications, Vol. 26, Springer-Verlag, New York Etc. 1992, Pp. 243–279. [REVIEW]James Cummings - 1995 - Journal of Symbolic Logic 60 (1):339-340.
  36.  5
    Review: Moti Gitik, Menachem Magidor, The Singular Cardinal Hypothesis Revisited. [REVIEW]James Cummings - 1995 - Journal of Symbolic Logic 60 (1):339-340.
  37.  5
    Review: Moti Gitik, The Strength of the Failure of the Singular Cardinal Hypothesis. [REVIEW]James Cummings - 1995 - Journal of Symbolic Logic 60 (1):340-340.
  38.  17
    Raymond M. Smullyan and Melvin Fitting. Set Theory and the Continuum Problem. Oxford Logic Guides, No. 34. Clarendon Press, Oxford University Press, Oxford, New York, Etc., 1996, Xiii + 288 Pp. [REVIEW]James Cummings - 1999 - Journal of Symbolic Logic 64 (1):401-403.
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  39.  18
    Review: Raymond M. Smullyan, Melvin Fitting, Set Theory and the Continuum Problem. [REVIEW]James Cummings - 1999 - Journal of Symbolic Logic 64 (1):401-403.
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  40.  18
    Review: Ten Papers by Arthur Apter on Large Cardinals. [REVIEW]James W. Cummings - 2000 - Bulletin of Symbolic Logic 6 (1):86 - 89.
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  41. Sheraton New Orleans Hotel, New Orleans, Louisiana January 12–13, 2001.James Cummings, Marcia Groszek & Dave Marker - 2001 - Bulletin of Symbolic Logic 7 (3).
     
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  42.  7
    The Hyper-Weak Distributive Law and a Related Game in Boolean Algebras.James Cummings & Natasha Dobrinen - 2007 - Annals of Pure and Applied Logic 149 (1-3):14-24.
    We discuss the relationship between various weak distributive laws and games in Boolean algebras. In the first part we give some game characterizations for certain forms of Prikry’s “hyper-weak distributive laws”, and in the second part we construct Suslin algebras in which neither player wins a certain hyper-weak distributivity game. We conclude that in the constructible universe L, all the distributivity games considered in this paper may be undetermined.
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  43.  3
    The Problem of Randomization Within a Standard of Care Range: A Case Study.James Cummings - 2015 - Journal of Clinical Research and Bioethics 6 (1).
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  44. The Tree Property at the Two Immediate Successors of a Singular Cardinal.James Cummings, Yair Hayut, Menachem Magidor, Itay Neeman, Dima Sinapova & Spencer Unger - forthcoming - Journal of Symbolic Logic:1-9.
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  45.  15
    2001 European Summer Meeting of the Association for Symbolic Logic Logic Colloquium'01.Itay Neeman, Alexander Leitsch, Toshiyasu Arai, Steve Awodey, James Cummings, Rod Downey & Harvey Friedman - 2002 - Bulletin of Symbolic Logic 8 (1):111-180.