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  1. Alexander Abian (1985). A Fixed Point Theorem Equivalent to the Axiom of Choice. Archive for Mathematical Logic 25 (1):173-174.
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  2. F. G. Abramson, L. A. Harrington, E. M. Kleinberg & W. S. Zwicker (1977). Flipping Properties: A Unifying Thread in the Theory of Large Cardinals. Annals of Mathematical Logic 12 (1):25-58.
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  3. W. Ackermann (1950). Tarski Alfred. Axiomatic and Algebraic Aspects of Two Theorems on Sums of Cardinals. Ebd., S. 79–104. Journal of Symbolic Logic 14 (4):257-258.
  4. Abraham Akkerman (1994). Sameness of Age Cohorts in the Mathematics of Population Growth. British Journal for the Philosophy of Science 45 (2):679-691.
    The axiom of extensionality of set theory states that any two classes that have identical members are identical. Yet the class of persons age i at time t and the class of persons age i + 1 at t + l, both including same persons, possess different demographic attributes, and thus appear to be two different classes. The contradiction could be resolved by making a clear distinction between age groups and cohorts. Cohort is a multitude of individuals, which is constituted (...)
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  5. Jesse Alama (2014). The Simplest Axiom System for Hyperbolic Geometry Revisited, Again. Studia Logica 102 (3):609-615.
    Dependencies are identified in two recently proposed first-order axiom systems for plane hyperbolic geometry. Since the dependencies do not specifically concern hyperbolic geometry, our results yield two simpler axiom systems for absolute geometry.
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  6. Maribel Anacona, Luis Carlos Arboleda & F. Javier Pérez-Fernández (2014). On Bourbaki's Axiomatic System for Set Theory. Synthese 191 (17):4069-4098.
    In this paper we study the axiomatic system proposed by Bourbaki for the Theory of Sets in the Éléments de Mathématique. We begin by examining the role played by the sign \(\uptau \) in the framework of its formal logical theory and then we show that the system of axioms for set theory is equivalent to Zermelo–Fraenkel system with the axiom of choice but without the axiom of foundation. Moreover, we study Grothendieck’s proposal of adding to Bourbaki’s system the axiom (...)
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  7. D. A. Anapolitanos & J. Väänänen (1980). On the Axiomatizability of the Notion of an Automorphism of a Finite Order. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 26 (28-30):433-437.
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  8. H. Andréka, Á Kurucz & I. Németi (1994). Connections Between Axioms of Set Theory and Basic Theorems of Universal Algebra. Journal of Symbolic Logic 59 (3):912-923.
    One of the basic theorems in universal algebra is Birkhoff's variety theorem: the smallest equationally axiomatizable class containing a class K of algebras coincides with the class obtained by taking homomorphic images of subalgebras of direct products of elements of K. G. Gratzer asked whether the variety theorem is equivalent to the Axiom of Choice. In 1980, two of the present authors proved that Birkhoff's theorem can already be derived in ZF. Surprisingly, the Axiom of Foundation plays a crucial role (...)
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  9. Arthur W. Apter (2016). Indestructibility and Destructible Measurable Cardinals. Archive for Mathematical Logic 55 (1-2):3-18.
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  10. 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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  11. 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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  12. M. Armbrust (1986). An Equivalence‐Theoretic Equivalent of the Axiom of Choice. Mathematical Logic Quarterly 32 (6):95-95.
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  13. M. Armbrust (1986). An Equivalence-Theoretic Equivalent of the Axiom of Choice. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 32 (6):95-95.
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  14. Roberto Arpaia (2005). Contributions to the History of the Axiom of Foundation. Epistemologia 28 (1).
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  15. Roberto Arpaia (2005). On the Negation of the Axiom of Foundation. Epistemologia 28 (2).
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  16. Wataru Asanuma, A Defense of Platonic Realism In Mathematics: Problems About The Axiom Of Choice.
    The conflict between Platonic realism and Constructivism marks a watershed in philosophy of mathematics. Among other things, the controversy over the Axiom of Choice is typical of the conflict. Platonists accept the Axiom of Choice, which allows a set consisting of the members resulting from infinitely many arbitrary choices, while Constructivists reject the Axiom of Choice and confine themselves to sets consisting of effectively specifiable members. Indeed there are seemingly unpleasant consequences of the Axiom of Choice. The non-constructive nature of (...)
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  17. Joan Bagaria (2002). Israel Journal of Mathematics. 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] Bulletin of Symbolic Logic 8 (4):543-545.
  18. 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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  19. Joan Bagaria & Menachem Magidor (2014). On ${\Omega _1}$-Strongly Compact Cardinals. Journal of Symbolic Logic 79 (1):266-278.
  20. Bernhard Banaschewski (1998). Choice Principles and Compactness Conditions. Mathematical Logic Quarterly 44 (3):427-430.
    It is shown in Zermelo-Fraenkel Set Theory that Cκ, the Axiom of Choice for κ-indexed families of arbitrary sets, is equivalent to the condition that the frame envelope of any κ-frame is κ-Lindelöf, for any cardinal κ.
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  21. Bernhard Banaschewski (1994). A New Proof That “Krull Implies Zorn”. Mathematical Logic Quarterly 40 (4):478-480.
    In the present note we give a direct deduction of the Axiom of Choice from the Maximal Ideal Theorem for commutative rings with unit.
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  22. Bernhard Banaschewski (1961). On Some Theorems Equivalent with the Axiom of Choice. Mathematical Logic Quarterly 7 (17‐18):279-282.
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  23. Bernhard Banaschewski (1961). On Some Theorems Equivalent with the Axiom of Choice. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 7 (17-18):279-282.
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  24. James E. Baumgartner (1975). Kleinberg E. M.. Strong Partition Properties for Infinite Cardinals. Journal of Symbolic Logic 40 (3):463.
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  25. James E. Baumgartner (1973). Addison J. W. And Moschovakis Yiannis N.. Some Consequences of the Axiom of Definable Determinateness. Proceedings of the National Academy of Sciences, Vol. 59 , Pp. 708–712.Martin Donald A.. The Axiom of Determinateness and Reduction Principles in the Analytical Hierarchy. Bulletin of the American Mathematical Society, Vol. 74 , Pp. 687–689. [REVIEW] Journal of Symbolic Logic 38 (2):334.
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  26. James E. Baumgartner (1971). Mycielski Jan and Steinhaus H.. A Mathematical Axiom Contradicting the Axiom of Choice. Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques Et Physiques, Vol. 10 , Pp. 1–3.Mycielski Jan. On the Axiom of Determinateness. Fundamenta Mathematicae, Vol. 53 , Pp. 205–224.Mycielski Jan and Świerczkowski S.. On the Lebesgue Measurability and the Axiom of Determinateness. Fundamenta Mathematicae, Vol. 54 , Pp. 67–71.Mycielski Jan. On the Axiom of Determinateness . Fundamenta Mathematicae, Vol. 59 , Pp. 203–212. [REVIEW] Journal of Symbolic Logic 36 (1):164-166.
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  27. Timothy Bays, Multi-Cardinal Phenomena in Stable Theories.
    In this dissertation we study two-cardinal phenomena—both of the admitting cardinals variety and of the Chang’s Conjecture variety—under the assumption that all our models have stable theories. All our results involve two, relatively widely accepted generalizations of the traditional definitions in this area. First, we allow the relevant subsets of our models to be picked out by (perhaps infinitary) partial types; second we consider δ-cardinal problems as well as two-cardinal problems.
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  28. Robert E. Beaudoin (1987). Strong Analogues of Martin's Axiom Imply Axiom R. Journal of Symbolic Logic 52 (1):216-218.
    We show that either PFA + or Martin's maximum implies Fleissner's Axiom R, a reflection principle for stationary subsets of P ℵ 1 (λ). In fact, the "plus version" (for one term denoting a stationary set) of Martin's axiom for countably closed partial orders implies Axiom R.
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  29. J. L. Bell (1974). On Compact Cardinals. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 20 (25-27):389-393.
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  30. John Bell, Measurable Cardinals.
    Let κ be an infinite cardinal. A κ-complete nonprincipal ultrafilter, or, for short, a κ- ultrafilter on a set A is a (nonempty) family U of subsets of A satisfying (i) S ⊆ U & |S|1 < κ ⇒ ∩S ∈ U (κ-completeness) (ii) X ∈ U & X ⊆ Y ⊆ A ⇒ Y ∈ U, (iii) ∀X ⊆ A [X ∈ U or A – X ∈ U] (iv) {a} ∉ U for any a..
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  31. John L. Bell (2009). The Axiom of Choice Vol. 22. College Publications.
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  32. Dorella Bellè & Franco Parlamento (2001). The Decidability of the Class and the Axiom of Foundation. Notre Dame Journal of Formal Logic 42 (1):41-53.
    We show that the Axiom of Foundation, as well as the Antifoundation Axiom AFA, plays a crucial role in determining the decidability of the following problem. Given a first-order theory T over the language , and a sentence F of the form with quantifier-free in the same language, are there models of T in which F is true? Furthermore we show that the Extensionality Axiom is quite irrelevant in that respect.
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  33. Omer Ben-Neria & Moti Gitik (2015). On the Splitting Number at Regular Cardinals. Journal of Symbolic Logic 80 (4):1348-1360.
  34. Itay Ben-Yaacov (2003). Discouraging Results for Ultraimaginary Independence Theory. Journal of Symbolic Logic 68 (3):846-850.
    Dividing independence for ultraimaginaries is neither symmetric nor transitive. Moreover, any notion of independence satisfying certain axioms (weaker than those for independence in a simple theory) and defined for all ultraimaginary sorts, is necessarily trivial.
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  35. Václav Edvard Beneš (1953). Rosser J. Barkley. The Axiom of Infinity in Quine's New Foundations. Journal of Symbolic Logic 18 (2):179.
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  36. Paul Bernays (1954). Aubert Karl Egil. Relations Généralisées Et Indépendance Logique des Notions de Réflexivité, Symétrie Et Transitivité. Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences , Vol. 229 , Pp. 538–540.Büchi J. Richard. Investigation of the Equivalence of the Axiom of Choice and Zorn's Lemma From the Viewpoint of the Hierarchy of Types. [REVIEW] Journal of Symbolic Logic 19 (4):285-286.
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  37. Andreas Blass (2005). Howard Paul and Rubin Jean E.. Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, Vol. 59. American Mathematical Society, Providence, RI, 1998, Viii+ 432 Pp. [REVIEW] Bulletin of Symbolic Logic 11 (1):61-63.
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  38. Will Boney (2014). Tameness From Large Cardinal Axioms. Journal of Symbolic Logic 79 (4):1092-1119.
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  39. Robert Earl Brandford (1971). Cardinal Addition and the Axiom of Choice. Annals of Mathematical Logic 3 (2):111-196.
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  40. Jorg Brendle (1991). Larger Cardinals in Cichon's Diagram. Journal of Symbolic Logic 56 (3):795.
  41. Norbert Brunner & Paul Howard (1992). Russell's Alternative to the Axiom of Choice. Mathematical Logic Quarterly 38 (1):529-534.
    We prove the independence of some weakenings of the axiom of choice related to the question if the unions of wellorderable families of wellordered sets are wellorderable.
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  42. Norbert Brunner & Paul Howard (1992). Russell's Alternative to the Axiom of Choice. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1):529-534.
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  43. Norbert Brunner, Karl Svozil & Matthias Baaz (1996). The Axiom of Choice in Quantum Theory. Mathematical Logic Quarterly 42 (1):319-340.
    We construct peculiar Hilbert spaces from counterexamples to the axiom of choice. We identify the intrinsically effective Hamiltonians with those observables of quantum theory which may coexist with such spaces. Here a self adjoint operator is intrinsically effective if and only if the Schrödinger equation of its generated semigroup is soluble by means of eigenfunction series expansions.
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  44. Everett L. Bull (1978). Successive Large Cardinals. Annals of Mathematical Logic 15 (2):161-191.
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  45. Timothy J. Carlson (2011). On the Conservativity of the Axiom of Choice Over Set Theory. Archive for Mathematical Logic 50 (7-8):777-790.
    We show that for various set theories T including ZF, T + AC is conservative over T for sentences of the form ${\forall x \exists ! y}$ A(x, y) where A(x, y) is a Δ0 formula.
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  46. Jesper Carlström (2004). EM + Ext− + ACint is Equivalent to ACext. Mathematical Logic Quarterly 50 (3):236-240.
    It is well known that the extensional axiom of choice implies the law of excluded middle . We here prove that the converse holds as well if we have the intensional axiom of choice ACint, which is provable in Martin-Löf's type theory, and a weak extensionality principle , which is provable in Martin-Löf's extensional type theory. In particular, EM is equivalent to ACext in extensional type theory.
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  47. 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.
  48. Yong Cheng & Victoria Gitman (2015). Indestructibility Properties of Remarkable Cardinals. Archive for Mathematical Logic 54 (7-8):961-984.
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  49. C. T. Chong (1982). Double Jumps of Minimal Degrees Over Cardinals. Journal of Symbolic Logic 47 (2):329-334.
  50. Alonzo Church (1950). Sobociński Bolesław. An Investigation of Protothetic. Cahiers de l'Institut d'Études Polonaises En Belgique, No. 5. Polycopié. Institut d'Etudes Polonaises En Belgique, Brussels 1949, V + 44 Pp. [REVIEW] Journal of Symbolic Logic 15 (1):64.
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