Search results for 'Continuum Hypothesis' (try it on Scholar)

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  1.  42
    Joel David Hamkins (2015). Is the Dream Solution of the Continuum Hypothesis Attainable? Notre Dame Journal of Formal Logic 56 (1):135-145.
    The dream solution of the continuum hypothesis would be a solution by which we settle the continuum hypothesis on the basis of a newly discovered fundamental principle of set theory, a missing axiom, widely regarded as true. Such a dream solution would indeed be a solution, since we would all accept the new axiom along with its consequences. In this article, however, I argue that such a dream solution to $\mathrm {CH}$ is unattainable.
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  2.  7
    Arthur L. Rubin & Jean E. Rubin (1993). Weak Forms of the Axiom of Choice and the Generalized Continuum Hypothesis. Mathematical Logic Quarterly 39 (1):7-22.
    In this paper we study some statements similar to the Partition Principle and the Trichotomy. We prove some relationships between these statements, the Axiom of Choice, and the Generalized Continuum Hypothesis. We also prove some independence results. MSC: 03E25, 03E50, 04A25, 04A50.
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  3.  15
    Paul J. Cohen (1966). Set Theory and the Continuum Hypothesis. New York, W. A. Benjamin.
  4.  13
    John-Michael Kuczynski (2016). The Power-Set Theorem and the Continuum Hypothesis: A Dialogue Concerning Infinite Number. Amazon Digital Services LLC.
    The nature of of Infinite Number is discussed in a rigorous but easy-to-follow manner. Special attention is paid to Cantor's proof that any given set has more subsets than members, and it is discussed how this fact bears on the question: How many infinite numbers are there? This work is ideal for people with little or no background in set theory who would like an introduction to the mathematics of the infinite.
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  5.  5
    Matthew Foreman & Menachem Magidor (1995). Large Cardinals and Definable Counterexamples to the Continuum Hypothesis. Annals of Pure and Applied Logic 76 (1):47-97.
    In this paper we consider whether has “enough information” to contain a counterexample to the continuum hypothesis. We believe this question provides deep insight into the difficulties surrounding the continuum hypothesis. We show sufficient conditions for not to contain such a counterexample. Along the way we establish many results about nonstationary towers, non-reflecting stationary sets, generalizations of proper and semiproper forcing and Chang's conjecture.
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  6. David J. Chalmers, Is the Continuum Hypothesis True, False, or Neither?
    Thanks to all the people who responded to my enquiry about the status of the Continuum Hypothesis. This is a really fascinating subject, which I could waste far too much time on. The following is a summary of some aspects of the feeling I got for the problems. This will be old hat to set theorists, and no doubt there are a couple of embarrassing misunderstandings, but it might be of some interest to non professionals.
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  7. Solomon Feferman, Is the Continuum Hypothesis a Definite Mathematical Problem?
    The purpose of this article is to explain why I believe that the Continuum Hypothesis (CH) is not a definite mathematical problem. My reason for that is that the concept of arbitrary set essential to its formulation is vague or underdetermined and there is no way to sharpen it without violating what it is supposed to be about. In addition, there is considerable circumstantial evidence to support the view that CH is not definite.
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  8.  37
    Gregory H. Moore (2011). Early History of the Generalized Continuum Hypothesis: 1878—1938. Bulletin of Symbolic Logic 17 (4):489-532.
    This paper explores how the Generalized Continuum Hypothesis (GCH) arose from Cantor's Continuum Hypothesis in the work of Peirce, Jourdain, Hausdorff, Tarski, and how GCH was used up to Gödel's relative consistency result.
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  9.  3
    Moti Gitik (1993). On Measurable Cardinals Violating the Continuum Hypothesis. Annals of Pure and Applied Logic 63 (3):227-240.
    Gitik, M., On measurable cardinals violating the continuum hypothesis, Annals of Pure and Applied Logic 63 227-240. It is shown that an extender used uncountably many times in an iteration is reconstructible. This together with the Weak Covering Lemma is used to show that the assumption o=κ+α is necessary for a measurable κ with 2κ=κ+α.
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  10.  58
    Thomas Weston (1976). Kreisel, the Continuum Hypothesis and Second Order Set Theory. Journal of Philosophical Logic 5 (2):281 - 298.
    The major point of contention among the philosophers and mathematicians who have written about the independence results for the continuum hypothesis (CH) and related questions in set theory has been the question of whether these results give reason to doubt that the independent statements have definite truth values. This paper concerns the views of G. Kreisel, who gives arguments based on second order logic that the CH does have a truth value. The view defended here is that although (...)
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  11. Jaakko Hintikka, Continuum Hypothesis as a Model-Theoretical Problem.
    Jaakko Hintikka 1. How to Study Set Theory The continuum hypothesis (CH) is crucial in the core area of set theory, viz. in the theory of the hierarchies of infinite cardinal and infinite ordinal numbers. It is crucial in that it would, if true, help to relate the two hierarchies to each other. It says that the second infinite cardinal number, which is known to be the cardinality of the first uncountable ordinal, equals the cardinality 2 o of (...)
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  12.  9
    Ramez L. Sami (1989). Turing Determinacy and the Continuum Hypothesis. Archive for Mathematical Logic 28 (3):149-154.
    From the hypothesis that all Turing closed games are determined we prove: (1) the Continuum Hypothesis and (2) every subset of ℵ1 is constructible from a real.
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  13.  7
    George C. Nelson (1998). Preservation Theorems Without Continuum Hypothesis. Studia Logica 60 (3):343-355.
    Many results concerning the equivalence between a syntactic form of formulas and a model theoretic conditions are proven directly without using any form of a continuum hypothesis. In particular, it is demonstrated that any reduced product sentence is equivalent to a Horn sentence. Moreover, in any first order language without equality one now has that a reduced product sentence is equivalent to a Horn sentence and any sentence is equivalent to a Boolean combination of Horn sentences.
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  14.  4
    Colin Rittberg (2015). How Woodin Changed His Mind: New Thoughts on the Continuum Hypothesis. Archive for History of Exact Sciences 69 (2):125-151.
    The Continuum Problem has inspired set theorists and philosophers since the days of Cantorian set theory. In the last 15 years, W. Hugh Woodin, a leading set theorist, has not only taken it upon himself to engage in this question, he has also changed his mind about the answer. This paper illustrates Woodin’s solutions to the problem, starting in Sect. 3 with his 1999–2004 argument that Cantor’s hypothesis about the continuum was incorrect. From 2010 onwards, Woodin presents (...)
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  15.  36
    Joseph Glicksohn (2004). Absorption, Hallucinations, and the Continuum Hypothesis. Behavioral and Brain Sciences 27 (6):793-794.
    The target article, in stressing the balance between neurobiological and psychological factors, makes a compelling argument in support of a continuum of perceptual and hallucinatory experience. Nevertheless, two points need to be addressed. First, the authors are probably underestimating the incidence of hallucinations in the normal population. Second, one should consider the role of absorption as a predisposing factor for hallucinations.
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  16.  8
    Kurt Gödel (1940). The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory. Princeton University Press;.
  17. W. Gielen, H. de Swart & W. Veldman (1981). The Continuum Hypothesis in Intuitionism. Journal of Symbolic Logic 46 (1):121-136.
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  18. Kurt Gödel (1940). The Consistency of the Continuum Hypothesis. Princeton University Press.
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  19.  18
    Paul Cohen (1963). The Independence of the Continuum Hypothesis. Proc. Nat. Acad. Sci. USA 50 (6):1143-1148.
  20.  6
    Paul J. Cohen (1963). The Independence of the Continuum Hypothesis. Proceedings of the National Academy of Sciences of the United States of America 50 (6):1143--8.
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  21.  14
    Frederick Bagemihl (1959). Some Results Connected with the Continuum Hypothesis. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 5 (7-13):97-116.
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  22.  14
    Stephen H. Hechler (1973). Powers of Singular Cardinals and a Strong Form of The Negation of The Generalized Continuum Hypothesis. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 19 (3-6):83-84.
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  23.  14
    Rolf Schock (1971). On the Axiom of Choice and the Continuum Hypothesis. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 17 (1):35-37.
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  24.  19
    Paul Cohen (1964). The Independence of the Continuum Hypothesis II. Proc. Nat. Acad. Sci. USA 51 (1):105-110.
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  25.  24
    John Gregory (1976). Higher Souslin Trees and the Generalized Continuum Hypothesis. Journal of Symbolic Logic 41 (3):663-671.
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  26.  12
    Roy O. Davies (1962). Equivalence to the Continuum Hypothesis of a Certain Proposition of Elementary Plane Geometry. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 8 (2):109-111.
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  27.  2
    Douglas S. Bridges (2016). The Continuum Hypothesis Implies Excluded Middle. In Peter Schuster & Dieter Probst (eds.), Concepts of Proof in Mathematics, Philosophy, and Computer Science. De Gruyter 111-114.
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  28.  16
    Frederick Bagemihl (1961). A Proposition of Elementary Plane Geometry That Implies the Continuum Hypothesis. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 7 (1-5):77-79.
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  29.  10
    Dana Scott (1968). A Proof of the Independence of the Continuum Hypothesis. Journal of Symbolic Logic 33 (2):293-293.
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  30.  15
    F. Bagemihl & S. Koo (1967). The Continuum Hypothesis and Ambiguous Points of Planar Functions. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 13 (13-14):219-223.
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  31.  34
    Uri Abraham (1983). On Forcing Without the Continuum Hypothesis. Journal of Symbolic Logic 48 (3):658-661.
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  32.  15
    Alexander Abian (1973). The Consistency of the Continuum Hypothesis Via Synergistic Models. Mathematical Logic Quarterly 19 (13):193-198.
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  33.  29
    Joel I. Friedman (1971). The Generalized Continuum Hypothesis is Equivalent to the Generalized Maximization Principle. Journal of Symbolic Logic 36 (1):39-54.
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  34.  25
    Thomas S. Weston (1977). The Continuum Hypothesis is Independent of Second-Order ZF. Notre Dame Journal of Formal Logic 18 (3):499-503.
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  35.  6
    W. Gielen, H. De Swart & W. Veldman (1981). The Continuum Hypothesis in Intuitionism. Journal of Symbolic Logic 46 (1):121 - 136.
  36.  1
    Natasha Dobrinen (2014). Cummings James and Schimmerling Ernest, Editors. Lecture Note Series of the London Mathematical Society, Vol. 406. Cambridge University Press, New York, Xi + 419 Pp.Larson Paul B., Lumsdaine Peter, and Yin Yimu. An Introduction to Pmax Forcing. Pp. 5–23.Thomas Simon and Schneider Scott. Countable Borel Equivalence Relations. Pp. 25–62.Farah Ilijas and Wofsey Eric. Set Theory and Operator Algebras. Pp. 63–119.Moore Justin and Milovich David. A Tutorial on Set Mapping Reflection. Pp. 121–144.Pestov Vladimir G. And Kwiatkowska Aleksandra. An Introduction to Hyperlinear and Sofic Groups. Pp. 145–185.Neeman Itay and Unger Spencer. Aronszajn Trees and the SCH. Pp. 187–206.Eisworth Todd, Tatch Moore Justin, and Milovich David. Iterated Forcing and the Continuum Hypothesis. Pp. 207–244.Gitik Moti and Unger Spencer. Short Extender Forcing. Pp. 245–263.Kechris Alexander S. And Tucker-Drob Robin D.. The Complexity of Classification Problems in Ergodic Theory. Pp. 265–299.Magidor Menachem and Lamb. [REVIEW] Bulletin of Symbolic Logic 20 (1):94-97.
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  37.  28
    Richard A. Platek (1969). Eliminating the Continuum Hypothesis. Journal of Symbolic Logic 34 (2):219-225.
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  38.  19
    Rolf Schock (1977). A Note on the Axiom of Choice and the Continuum Hypothesis. Notre Dame Journal of Formal Logic 18 (3):409-414.
  39.  15
    Raouf Doss (1963). On Gödel's Proof That $V=L$ Implies the Generalized Continuum Hypothesis. Notre Dame Journal of Formal Logic 4 (4):283-287.
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  40.  6
    Roy O. Davies (1962). Equivalence to the Continuum Hypothesis of a Certain Proposition of Elementary Plane Geometry. Mathematical Logic Quarterly 8 (2):109-111.
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  41.  1
    Kenneth Kunen (1969). Vopěnka P.. The Limits of Sheaves and Applications on Constructions of Models. Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques Et Physiques, Vol. 13 , Pp. 189–192.Vopěnka P.. On ∇-Model of Set Theory. Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques Et Physiques, Vol. 13 , Pp. 267–272.Vopěnka P.. Properties of ∇-Model. Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques Et Physiques, Vol. 13 , Pp. 441–444.Vopěnka P. And Hájek P.. Permutation Submodels of the Model ∇. Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques Et Physiques, Vol. 13 , Pp. 611–614.Hájek P. And Vopěnka P.. Some Permutation Submodels of the Model ∇. Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques Et Physiques, Vol. 14 , Pp. 1–7.Vopěnka P.. ∇-Models in Which the Generalized Continuum Hypothesis. [REVIEW] Journal of Symbolic Logic 34 (3):515-516.
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  42.  1
    Leslie H. Tharp (1970). Karp Carol. A Proof of the Relative Consistency of the Continuum Hypothesis. Sets, Models and Recursion Theory, Proceedings of the Summer School in Mathematical Logic and Tenth Logic Colloquium, Leicester, August-September 1965, Edited by Crossley John N., Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam, and Humanities Press, New York, 1967, Pp. 1–32. [REVIEW] Journal of Symbolic Logic 35 (2):344-345.
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  43.  5
    Boleslaw Sobocinski (1963). A Note On The Generalized Continuum Hypothesis, Ii. Notre Dame Journal of Formal Logic 4 (1):67-79.
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  44.  7
    Bolesław Sobociński (1963). A Note on the Generalized Continuum Hypothesis. II. Notre Dame Journal of Formal Logic 4 (1):67-79.
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  45.  4
    Stephen H. Hechler (1973). Powers of Singular Cardinals and a Strong Form of The Negation of The Generalized Continuum Hypothesis. Mathematical Logic Quarterly 19 (3‐6):83-84.
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  46.  4
    F. R. Drake (1973). Review: L. Rieger, On the Consistency of the Generalized Continuum Hypothesis. [REVIEW] Journal of Symbolic Logic 38 (1):153-153.
  47.  8
    Bolesław Sobociński (1962). A Note on the Generalized Continuum Hypothesis. I. Notre Dame Journal of Formal Logic 3 (4):274-278.
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  48.  6
    Bolesław Sobociński (1963). A Note on the Generalized Continuum Hypothesis. III. Notre Dame Journal of Formal Logic 4 (3):233-240.
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  49. Syd Friedman & Peter Koepke (1997). X1. Introduction. In 1938, K. Gödel Defined the Model L of Set Theory to Show the Relative Consistency of Cantor's Continuum Hypothesis. L is Defined as a Union L=. [REVIEW] Bulletin of Symbolic Logic 3 (4).
     
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  50.  3
    Paul Bernays (1940). Review: Kurt Godel, Consistency-Proof for the Generalized Continuum-Hypothesis. [REVIEW] Journal of Symbolic Logic 5 (3):117-118.
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