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  1. Arthur W. Apter (2016). Indestructibility and Destructible Measurable Cardinals. Archive for Mathematical Logic 55 (1-2):3-18.
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  2. Arthur W. Apter (2009). Indestructibility and Stationary Reflection. Mathematical Logic Quarterly 55 (3):228-236.
    If κ < λ are such that κ is a strong cardinal whose strongness is indestructible under κ -strategically closed forcing and λ is weakly compact, then we show thatA = {δ < κ | δ is a non-weakly compact Mahlo cardinal which reflects stationary sets}must be unbounded in κ. This phenomenon, however, need not occur in a universe with relatively few large cardinals. In particular, we show how to construct a model where no cardinal is supercompact up to a (...)
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  3. Arthur W. Apter (2009). Indestructibility Under Adding Cohen Subsets and Level by Level Equivalence. Mathematical Logic Quarterly 55 (3):271-279.
    We construct a model for the level by level equivalence between strong compactness and supercompactness in which the least supercompact cardinal κ has its strong compactness indestructible under adding arbitrarily many Cohen subsets. There are no restrictions on the large cardinal structure of our model.
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  4. Joan Bagaria, Joel David Hamkins, Konstantinos Tsaprounis & Toshimichi Usuba (2016). Superstrong and Other Large Cardinals Are Never Laver Indestructible. Archive for Mathematical Logic 55 (1-2):19-35.
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  5. Frederick Bagemihl (1959). Some Results Connected with the Continuum Hypothesis. Mathematical Logic Quarterly 5 (7‐13):97-116.
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  6. 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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  7. J. L. Bell (1974). On Compact Cardinals. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 20 (25-27):389-393.
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  8. Everett L. Bull (1978). Successive Large Cardinals. Annals of Mathematical Logic 15 (2):161-191.
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  9. Yong Cheng, Sy-David Friedman & Joel David Hamkins (2015). Large Cardinals Need Not Be Large in HOD. Annals of Pure and Applied Logic 166 (11):1186-1198.
  10. Yong Cheng & Victoria Gitman (2015). Indestructibility Properties of Remarkable Cardinals. Archive for Mathematical Logic 54 (7-8):961-984.
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  11. Paul E. Cohen (1974). L-Mahlo Cardinals. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 20 (13-18):229-231.
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  12. 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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  13. James Cummings, Sy David Friedman & Mohammad Golshani (2015). Collapsing the Cardinals of HOD. Journal of Mathematical Logic 15 (2):1550007.
  14. 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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  15. 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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  16. 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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  17. Sy D. Friedman (2005). Genericity and Large Cardinals. Journal of Mathematical Logic 5 (02):149-166.
  18. 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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  19. Sy-David Friedman & Philipp Lücke (2015). Large Cardinals and Definable Well-Orders, Without the GCH. Annals of Pure and Applied Logic 166 (3):306-324.
  20. 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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  21. P. K. H. (1967). Set Theory and the Continuum Hypothesis. [REVIEW] Review of Metaphysics 20 (4):716-716.
  22. 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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  23. 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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  24. 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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  25. J. M. Henle & E. M. Kleinberg (1978). A Flipping Characterization of Ramsey Cardinals. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 24 (1-6):31-36.
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  26. John L. Hickman (1979). Boundedness Properties of Cardinals. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 25 (31):485-486.
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  27. Paul Howard & Eleftherios Tachtsis (forthcoming). No Decreasing Sequence of Cardinals. Archive for Mathematical Logic.
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  28. 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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  29. C. A. Johnson (1986). Precipitous Ideals on Singular Cardinals. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 32 (25-30):461-465.
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  30. Akihiro Kanamori (1994). The Higher Infinite Large Cardinals in Set Theory From Their Beginnings.
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  31. E. M. Kleinberg (1974). A Combinatorial Property of Measurable Cardinals. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 20 (7):109-111.
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  32. 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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  33. Azriel Levy (1978). Review: Frank R. Drake, Set Theory. An Introduction to Large Cardinals. [REVIEW] Journal of Symbolic Logic 43 (2):384-384.
  34. R. Lubarsky (2008). R. Taschner, The Continuum. Bulletin of Symbolic Logic 14 (2).
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  35. Itay Neeman & John Steel (2016). Equiconsistencies at Subcompact Cardinals. Archive for Mathematical Logic 55 (1-2):207-238.
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  36. 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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  37. 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.
    (No abstract is available for this citation).
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  38. Manuel Rebuschi (1996). Sur le paradoxe dit «de Burali-Forti». Philosophia Scientiae 1 (1):111-124.
    L'historiographie des paradoxes ensemblistes attribue classiquement la "découverte" du paradoxe du plus grand ordinal au mathématicien italien Burali-Forti. Un examen attentif de ses démonstration et revirement révèle qu'il n'en est rien. Nous tenterons de dégager l'impact de la publication des Principles de Russell sur la constitution de cette version historiographique officielle, avant d'aborder la question de la définition des paradoxes et ce qui peut en motiver une conception restrictive.
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  39. 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 a very different (...)
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  40. Rudolf Taschner & Robert Lubarsky (2008). The Continuum. Bulletin of Symbolic Logic 14 (2):260-261.
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  41. Nam Trang (2015). Derived Models and Supercompact Measures on ℘Ω1). Mathematical Logic Quarterly 61 (1-2):56-65.
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  42. José P. Úbeda (1975). Frank R. Drake," Set Theory: An Introduction to Large Cardinals". Teorema: International Journal of Philosophy 5 (3):521-525.
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  43. 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. Edward G. Belaga (forthcoming). Retrieving the Mathematical Mission of the Continuum Concept From the Transfinitely Reductionist Debris of Cantor’s Paradise. Extended Abstract. International Journal of Pure and Applied Mathematics.
    What is so special and mysterious about the Continuum, this ancient, always topical, and alongside the concept of integers, most intuitively transparent and omnipresent conceptual and formal medium for mathematical constructions and the battle field of mathematical inquiries ? And why it resists the century long siege by best mathematical minds of all times committed to penetrate once and for all its set-theoretical enigma ? -/- The double-edged purpose of the present study is to save from the transfinite deadlock of (...)
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  5. L. Bell John (2011). Set Theory: Boolean-Valued Models and Independence Proofs. Oxford University Press.
    This third edition, now available in paperback, is a follow up to the author's classic Boolean-Valued Models and Independence Proofs in Set Theory. It provides an exposition of some of the most important results in set theory obtained in the 20th century: the independence of the continuum hypothesis and the axiom of choice.
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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. 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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