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  1. Chance and the Continuum Hypothesis.Daniel Hoek - forthcoming - Philosophy and Phenomenological Research.
    This paper presents and defends an argument that the continuum hypothesis is false, based on considerations about objective chance and an old theorem due to Banach and Kuratowski. More specifically, I argue that the probabilistic inductive methods standardly used in science presuppose that every proposition about the outcome of a chancy process has a certain chance between 0 and 1. I also argue in favour of the standard view that chances are countably additive. Since it is possible to randomly pick (...)
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  2. H. J. Keisler and A. Tarski. From Accessible to Inaccessible Cardinals. Fundamenta Mathematicae, Vol. 53 , Pp. 225–308. , P. 119.). [REVIEW]Azriel Lévy - 1967 - Journal of Symbolic Logic 32 (3):411.
  3. Steven Orey. New Foundations and the Axiom of Counting. Duke Mathematical Journal, Vol. 31 , Pp. 655–660.Norman Feldman - 1969 - Journal of Symbolic Logic 34 (4):649.
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  4. A. Lévy and R. M. Solovay. Measurable Cardinals and the Continuum Hypothesis. Israel Journal of Mathematics, Vol. 5 , Pp. 234–248. [REVIEW]F. R. Drake - 1969 - Journal of Symbolic Logic 34 (4):654-655.
  5. Takeo Sugihara. The Numbers of Modalities in T Supplemented by the Axiom CL2pL3p. The Journal of Symbolic Logic, Vol. 27 No. 4 , Pp. 407–408.Krister Segerberg - 1969 - Journal of Symbolic Logic 34 (2):305.
  6. C. C. Chang. Maximal N-Disjointed Sets and the Axiom of Choice. Fundamenta Mathematicae, Vol. 49 , Pp. 11–14.Azriel Lévy - 1970 - Journal of Symbolic Logic 35 (3):473.
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  7. J. W. Addison and Yiannis N. Moschovakis. Some Consequences of the Axiom of Definable Determinateness. Proceedings of the National Academy of Sciences, Vol. 59 , Pp. 708–712. - Donald A. Martin. The Axiom of Determinateness and Reduction Principles in the Analytical Hierarchy. Bulletin of the American Mathematical Society, Vol. 74 , Pp. 687–689. [REVIEW]James E. Baumgartner - 1973 - Journal of Symbolic Logic 38 (2):334.
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  8. Shaligram Singh. The Independence of a Strong Axiom of Choice. The Mathematical Gazette, Vol. 46 , Pp. 126–129.H. B. Enderton - 1973 - Journal of Symbolic Logic 38 (2):335.
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  9. E. M. Kleinberg. Strong Partition Properties for Infinite Cardinals. The Journal of Symbolic Logic, Vol. 35 , Pp. 410–428.James E. Baumgartner - 1975 - Journal of Symbolic Logic 40 (3):463.
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  10. Jack H. Silver. Measurable Cardinals and Well-Orderings. Annals of Mathematics, Ser. 2 Vol. 94 , Pp. 414–446.Menachem Magidor - 1974 - Journal of Symbolic Logic 39 (2):330-331.
  11. Herman Rubin and Jean E. Rubin. Equivalents of the Axiom of Choice, II. Studies in Logic and the Foundations of Mathematics, Vol. 116. North-Holland, Amsterdam, New York, and Oxford, 1985, Xxviii + 322 Pp. [REVIEW]David Pincus - 1987 - Journal of Symbolic Logic 52 (3):867-869.
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  12. Donald A. Martin and John R. Steel. Projective Determinacy. Proceedings of the National Academy of Sciences of the United States of America, Vol. 85 , Pp. 6582–6586. - W. Hugh Woodin. Supercompact Cardinals, Sets of Reals, and Weakly Homogeneous Trees. Proceedings of the National Academy of Sciences of the United States of America, Vol. 85 , Pp. 6587–6591. - Donald A. Martin and John R. Steel. A Proof of Projective Determinacy. Journal of the American Mathematical Society, Vol. 2 , Pp. 71–125. [REVIEW]Matthew D. Foreman - 1992 - Journal of Symbolic Logic 57 (3):1132-1136.
  13. Michiel van Lambalgen. Independence, Randomness and the Axiom of Choice. The Journal of Symbolic Logic, Vol. 57 , Pp. 1274–1304.John C. Simms - 1994 - Journal of Symbolic Logic 59 (4):1433-1434.
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  14. 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.
  15. William Mitchell, Ernest Schimmerling, and John Steel. The Covering Lemma Up to a Woodin Cardinal. Annals of Pure and Applied Logic, Vol. 84 , Pp. 219–255. [REVIEW]Itay Neeman - 2003 - Bulletin of Symbolic Logic 9 (3):414-416.
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  16. Saharon Shelah and Hugh Woodin. Large Cardinals Imply That Every Reasonably Definable Set of Reals is Lebesgue Measurable. Israel Journal of Mathematics, Vol. 70 , Pp. 381–394. [REVIEW]Joan Bagaria - 2002 - Bulletin of Symbolic Logic 8 (4):543-545.
  17. W. Hugh Woodin. The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. De Gruyter Series in Logic and its Applications, No. 1. Walter de Gruyter, Berlin and New York 1999, Vi + 934 Pp. [REVIEW]Paul Larson - 2002 - Bulletin of Symbolic Logic 8 (1):91-93.
  18. Paul Howard and Jean E. Rubin. Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, Vol. 59. American Mathematical Society, Providence, RI, 1998, Viii + 432 Pp. [REVIEW]Andreas Blass - 2005 - Bulletin of Symbolic Logic 11 (1):61-63.
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  19. Constructive Set Theory with Operations.Andrea Cantini & Laura Crosilla - 2008 - In Logic Colloquium 2004.
    We present an extension of constructive Zermelo{Fraenkel set theory [2]. Constructive sets are endowed with an applicative structure, which allows us to express several set theoretic constructs uniformly and explicitly. From the proof theoretic point of view, the addition is shown to be conservative. In particular, we single out a theory of constructive sets with operations which has the same strength as Peano arithmetic.
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  20. No Decreasing Sequence of Cardinals.Paul Howard & Eleftherios Tachtsis - 2016 - Archive for Mathematical Logic 55 (3-4):415-429.
    In set theory without the Axiom of Choice, we investigate the set-theoretic strength of the principle NDS which states that there is no function f on the set ω of natural numbers such that for everyn ∈ ω, f ≺ f, where for sets x and y, x ≺ y means that there is a one-to-one map g : x → y, but no one-to-one map h : y → x. It is a long standing open problem whether NDS implies (...)
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  21. Modal Set Theory.Christopher Menzel - forthcoming - In Otávio Bueno & Scott Shalkowski (eds.), The Routledge Handbook of Modality. London and New York: Routledge.
    This article presents an overview of the basic philosophical motivations for, and some recent work in, modal set theory.
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  22. Characterizing Large Cardinals in Terms of Layered Posets.Sean Cox & Philipp Lücke - 2017 - Annals of Pure and Applied Logic 168 (5):1112-1131.
  23. On the Theory of Axiom-Systems.Olaf Helmer - 1935 - Analysis 3 (1-2):1-11.
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  24. Epistemology of Logic - Logic-Dialectic or Theory of the Knowledge.Epameinondas Xenopoulos (ed.) - 1998 - Kefalonia,GREECE: KATERINA XENOPOULOU.
    1994.Επιστημολογία της Λογικής. Συγγραφέας Επαμεινώνδας Ξενόπουλος Μοναδική μελέτη και προσέγγιση της θεωρίας της γνώσης, για την παγκόσμια βιβλιογραφία, της διαλεκτικής πορείας της σκέψης από την λογική πλευρά της και της μελλοντικής μορφής που θα πάρουν οι διαλεκτικές δομές της, στην αδιαίρετη ενότητα γνωσιοθεωρίας, λογικής και διαλεκτικής, με την «μέθοδο του διαλεκτικού υλισμού». Έργο βαρύ με θέμα εξαιρετικά δύσκολο διακατέχεται από πρωτοτυπία και ζωντάνια που γοητεύει τον κάθε ανήσυχο στοχαστή από τις πρώτες γραμμές.
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  25. Games, Scales, and Suslin Cardinals. The Cabal Seminar, Volume I, Lecture Notes in Logic, Vol. 31.Alessandro Andretta - 2012 - Bulletin of Symbolic Logic 18 (1):122-126.
  26. AD[Syntactic Turnstile] the [Aleph]"N" Are Jonsson Cardinals and [Aleph] Omega is a Rowbottom Cardinal.E. M. Kleinberg - 1977 - Annals of Mathematical Logic 12 (3):229.
  27. The Necessary Maximality Principle for C. C. C. Forcing is Equiconsistent with a Weakly Compact Cardinal.Joel D. Hamkins & W. Hugh Woodin - 2005 - Mathematical Logic Quarterly 51 (5):493-498.
    The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal.
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  28. Some Results on Partitions and Cartesian Products in the Absence of the Axiom of Choice.A. H. Kruse - 1974 - Mathematical Logic Quarterly 20 (8-12):149-172.
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  29. Concerning the Proper Axioms of S4.02.Bolesław Sobociński - 1974 - Notre Dame Journal of Formal Logic 15:169.
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  30. The Cardinals Below [Ω1]<Ω1.W. Hugh Woodin - 2006 - Annals of Pure and Applied Logic 140 (1-3):161-232.
    The results of this paper concern the effective cardinal structure of the subsets of [ω1]<ω1, the set of all countable subsets of ω1. The main results include dichotomy theorems and theorems which show that the effective cardinal structure is complicated.
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  31. Boolean Extensions and Measurable Cardinals.K. Kunen - 1971 - Annals of Mathematical Logic 2 (4):359.
  32. Powers of Regular Cardinals.William B. Easton - 1970 - Annals of Mathematical Logic 1 (2):139.
  33. Successive Large Cardinals.Everett L. Bull - 1978 - Annals of Mathematical Logic 15 (2):161.
  34. Omega ¹-Constructible Universe and Measurable Cardinals.Claude Sureson - 1986 - Annals of Pure and Applied Logic 30 (3):293.
  35. Strong Compactness and Other Cardinal Sins.Jussi Ketonen - 1972 - Annals of Mathematical Logic 5 (1):47.
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  36. Some Weak Versions of Large Cardinal Axioms.Keith J. Devlin - 1973 - Annals of Mathematical Logic 5 (4):291.
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  37. Some Combinatorial Problems Concerning Uncountable Cardinals.Thomas J. Jech - 1973 - Annals of Mathematical Logic 5 (3):165.
  38. Adding Closed Cofinal Sequences to Large Cardinals.Lon Berk Radin - 1982 - Annals of Mathematical Logic 22 (3):243.
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  39. Flipping Properties: A Unifying Thread in the Theory of Large Cardinals.F. G. Abramson, L. A. Harrington, E. M. Kleinberg & W. S. Zwicker - 1977 - Annals of Mathematical Logic 12 (1):25.
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  40. How Many Normal Measures Can ℵmath Image Carry?Arthur W. Apter - 2010 - Mathematical Logic Quarterly 56 (2):164-170.
    Relative to the existence of a supercompact cardinal with a measurable cardinal above it, we show that it is consistent for ℵ1 to be regular and for ℵmath image to be measurable and to carry precisely τ normal measures, where τ ≥ ℵmath image is any regular cardinal. This extends the work of [2], in which the analogous result was obtained for ℵω +1 using the same hypotheses.
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  41. Sequential Topological Conditions in ℝ in the Absence of the Axiom of Choice.Gonçalo Gutierres - 2003 - Mathematical Logic Quarterly 49 (3):293-298.
    It is known that – assuming the axiom of choice – for subsets A of ℝ the following hold: A is compact iff it is sequentially compact, A is complete iff it is closed in ℝ, ℝ is a sequential space. We will show that these assertions are not provable in the absence of the axiom of choice, and that they are equivalent to each.
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  42. On Κ-Hereditary Sets and Consequences of the Axiom of Choice.Karl-Heinz Diener - 2000 - Mathematical Logic Quarterly 46 (4):563-568.
    We will prove that some so-called union theorems are equivalent in ZF0 to statements about the transitive closure of relations. The special case of “bounded” union theorems dealing with κ-hereditary sets yields equivalents to statements about the transitive closure of κ-narrow relations. The instance κ = ω1 yields an equivalent to Howard-Rubin's Form 172 of every hereditarily countable set x is countable). In particular, the countable union theorem and, a fortiori, the axiom of countable choice imply Form 172.
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  43. Compact and Loeb Hausdorff Spaces in Equation Image and the Axiom of Choice for Families of Finite Sets.Kyriakos Keremedis - 2012 - Mathematical Logic Quarterly 58 (3):130-138.
    Given a set X, equation image denotes the statement: “equation image has a choice set” and equation image denotes the family of all closed subsets of the topological space equation image whose definition depends on a finite subset of X. We study the interrelations between the statements equation image equation image equation image equation image and “equation imagehas a choice set”. We show: equation image iff equation image iff equation image has a choice set iff equation image. equation image iff (...)
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  44. Axiom of Choice and Complementation.Radu Diaconescu - 1975 - Proceedings of the American Mathematical Society 51 (1):176-178.
    It is shown that an intuitionistic model of set theory with the axiom of choice has to be a classical one.
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  45. Assertion Proof and the Axiom of Choice.David DeVidi - 2006 - In ¸ Itedevidikenyon2006. Springer Verlag.
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  46. The Axiom of Choice Vol. 22.John L. Bell - 2009 - College Publications.
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  47. Pantachies and Weakly Inaccessible Cardinals.Koji Nakatogawa - 1987 - Annals of the Japan Association for Philosophy of Science 7 (2):57-71.
  48. On ^|^Alefsym;0-Complete Cardinals and ^|^Pi;11-Class of Ordinals.Kanji Namba - 1967 - Annals of the Japan Association for Philosophy of Science 3 (2):77-86.
  49. Criteria of Identity and the Axiom of Choice.Timothy Williamson - 1986 - Journal of Philosophy 83 (7):380.
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  50. A Generalized Cut Characterization of the Fullness Axiom in CZF.Laura Crosilla, Erik Palmgren & Peter Schuster - 2013 - Logic Journal of the IGPL 21 (1):63-76.
    In the present note, we study a generalization of Dedekind cuts in the context of constructive Zermelo–Fraenkel set theory CZF. For this purpose, we single out an equivalent of CZF's axiom of fullness and show that it is sufficient to derive that the Dedekind cuts in this generalized sense form a set. We also discuss the instance of this equivalent of fullness that is tantamount to the assertion that the class of Dedekind cuts in the rational numbers, in the customary (...)
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