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  1. Frederick Bagemihl (1959). Some Results Connected with the Continuum Hypothesis. Mathematical Logic Quarterly 5 (7‐13):97-116.
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  2. Everett L. Bull (1978). Successive Large Cardinals. Annals of Mathematical Logic 15 (2):161-191.
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  3. Paul Corazza (2010). The Axiom of Infinity and Transformations J: V→V. Bulletin of Symbolic Logic 16 (1):37-84.
    We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be derivable? (...)
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  4. Keith J. Devlin (1980). Concerning the Consistency of the Souslin Hypothesis with the Continuum Hypothesis. Annals of Mathematical Logic 19 (1-2):115-125.
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  5. F. R. Drake (1973). Review: L. Rieger, On the Consistency of the Generalized Continuum Hypothesis. [REVIEW] Journal of Symbolic Logic 38 (1):153-153.
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  6. Sy D. Friedman (2005). Definability Degrees. Mathematical Logic Quarterly 51 (5):448-449.
    We establish the equiconsistency of a simple statement in definability theory with the failure of the GCH at all infinite cardinals. The latter was shown by Foreman and Woodin to be consistent, relative to the existence of large cardinals.
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  7. Sy D. Friedman (2005). Genericity and Large Cardinals. Journal of Mathematical Logic 5 (02):149-166.
  8. Sy D. Friedman (2003). Cardinal-Preserving Extensions. Journal of Symbolic Logic 68 (4):1163-1170.
    A classic result of Baumgartner-Harrington-Kleinberg [1] implies that assuming CH a stationary subset of ω1 has a CUB subset in a cardinal-perserving generic extension of V, via a forcing of cardinality ω1. Therefore, assuming that $\omega_2^L$ is countable: { $X \in L \mid X \subseteq \omega_1^L$ and X has a CUB subset in a cardinal -preserving extension of L} is constructible, as it equals the set of constructible subsets of $\omega_1^L$ which in L are stationary. Is there a similar such (...)
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  9. 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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  10. 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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  11. Michał Heller & W. H. Woodin (eds.) (2011). Infinity: New Research Frontiers. Cambridge University Press.
    Machine generated contents note: Introduction Rudy Rucker; Part I. Perspectives on Infinity from History: 1. Infinity as a transformative concept in science and theology Wolfgang Achtner; Part II. Perspectives on Infinity from Mathematics: 2. The mathematical infinity Enrico Bombieri; 3. Warning signs of a possible collapse of contemporary mathematics Edward Nelson; Part III. Technical Perspectives on Infinity from Advanced Mathematics: 4. The realm of the infinite W. Hugh Woodin; 5. A potential subtlety concerning the distinction between determinism and nondeterminism W. (...)
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  12. Ignasi Jané (2010). Idealist and Realist Elements in Cantor's Approach to Set Theory. Philosophia Mathematica 18 (2):193-226.
    There is an apparent tension between the open-ended aspect of the ordinal sequence and the assumption that the set-theoretical universe is fully determinate. This tension is already present in Cantor, who stressed the incompletable character of the transfinite number sequence in Grundlagen and avowed the definiteness of the totality of sets and numbers in subsequent philosophical publications and in correspondence. The tension is particularly discernible in his late distinction between sets and inconsistent multiplicities. I discuss Cantor’s contrasting views, and I (...)
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  13. I. Jane (2010). Idealist and Realist Elements in Cantor's Approach to Set Theory. Philosophia Mathematica 18 (2):193-226.
    There is an apparent tension between the open-ended aspect of the ordinal sequence and the assumption that the set-theoretical universe is fully determinate. This tension is already present in Cantor, who stressed the incompletable character of the transfinite number sequence in Grundlagen and avowed the definiteness of the totality of sets and numbers in subsequent philosophical publications and in correspondence. The tension is particularly discernible in his late distinction between sets and inconsistent multiplicities. I discuss Cantor’s contrasting views, and I (...)
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  14. Azriel Levy (1978). Review: Frank R. Drake, Set Theory. An Introduction to Large Cardinals. [REVIEW] Journal of Symbolic Logic 43 (2):384-384.
  15. R. Lubarsky (2008). R. Taschner, The Continuum. Bulletin of Symbolic Logic 14 (2).
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  16. Matthew W. Parker (2013). Set Size and the Part-Whole Principle. Review of Symbolic Logic (4):1-24.
    Recent work has defended “Euclidean” theories of set size, in which Cantor’s Principle (two sets have equally many elements if and only if there is a one-to-one correspondence between them) is abandoned in favor of the Part-Whole Principle (if A is a proper subset of B then A is smaller than B). It has also been suggested that Gödel’s argument for the unique correctness of Cantor’s Principle is inadequate. Here we see from simple examples, not that Euclidean theories of set (...)
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  17. Esther Ramharter (2009). Christine Redecker. Wittgensteins Philosophie der Mathematik: Eine Neubewertung Im Ausgang Von der Kritik an Cantors Beweis der Überabzählbarkeit der Reellen Zahlen. [Wittgenstein's Philosophy of Mathematics: A Reassessment Starting From the Critique of Cantor's Proof of the Uncountability of the Real Numbers]. Philosophia Mathematica 17 (3):382-392.
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  18. Rudolf Taschner & Robert Lubarsky (2008). The Continuum. Bulletin of Symbolic Logic 14 (2):260-261.
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  19. José P. Úbeda (1975). Frank R. Drake," Set Theory: An Introduction to Large Cardinals". Teorema: Revista Internacional de Filosofía 5 (3):521-525.
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  20. Gabriel Uzquiano (2005). Review of M. Potter, Set Theory and its Philosophy: A Critical Introduction. [REVIEW] Philosophia Mathematica 13 (3):308-346.
The Continuum Hypothesis
  1. Uri Abraham (1983). On Forcing Without the Continuum Hypothesis. Journal of Symbolic Logic 48 (3):658-661.
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  2. C. Alvarez Jimenez (1995). Some Logical Remarks Concerning the Continuum Problem. Boston Studies in the Philosophy of Science 172:173-186.
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  3. Arthur W. Apter (2002). Review of J. Cummings, A Model in Which GCH Holds at Successors but Fails at Limits; Strong Ultrapowers and Long Core Models; Coherent Sequences Versus Radin Sequences; and J. Cummings, M. Foreman, and M. Magidor, Squares, Scales and Stationary Reflection. [REVIEW] Bulletin of Symbolic Logic 8 (4):550-552.
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  4. 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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  5. Justin Clarke-Doane (2013). What is Absolute Undecidability?†. Noûs 47 (3):467-481.
    It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical (...)
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  6. Paul J. Cohen (1966). Set Theory and the Continuum Hypothesis. New York, W. A. Benjamin.
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  7. 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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  8. 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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  9. Solomon Feferman, Conceptions of the Continuum.
    Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions.
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  10. Solomon Feferman, Conceptual Structuralism and the Continuum.
    • This comes from my general view of the nature of mathematics, that it is humanly based and that it deals with more or less clear conceptions of mathematical structures; for want of a better word, I call that view conceptual structuralism.
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  11. 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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  12. 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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  13. 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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  14. 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;.
  15. Kurt Gödel (1940). The Consistency of the Continuum Hypothesis. Princeton University Press.
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  16. John Gregory (1976). Higher Souslin Trees and the Generalized Continuum Hypothesis. Journal of Symbolic Logic 41 (3):663-671.
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  17. 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 the continuum. (...)
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  18. 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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  19. Anne Newstead & Franklin James (2008). On the Reality of the Continuum. Philosophy 83 (01):117-28.
    In a recent article (‘The Continuum: Russell’s Moment of Candour’), Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably infinite set. He rejects the conclusion of Cantor’s diagonal argument for the higher, non-denumerable infinity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only ‘well-defined’ real numbers as proper objects of study. (...)
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  20. Richard A. Platek (1969). Eliminating the Continuum Hypothesis. Journal of Symbolic Logic 34 (2):219-225.
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  21. Assaf Rinot (2010). The Search for Diamonds: Review of S. Shelah, Middle Diamond; S. Shelah, Diamonds; and M. Zeman, Diamond, GCH and Weak Square. [REVIEW] Bulletin of Symbolic Logic 16 (3):420 - 423.
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  22. Rolf Schock (1977). A Note on the Axiom of Choice and the Continuum Hypothesis. Notre Dame Journal of Formal Logic 18 (3):409-414.
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  23. Rolf Schock (1966). A Simple Version of the Generalized Continuum Hypothesis. Notre Dame Journal of Formal Logic 7 (3):287-288.
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  24. 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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  25. 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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  26. 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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  27. 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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  28. 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 Kreisel's conclusion (...)
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  29. 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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