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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. 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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  4. 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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  5. 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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  6. M. Armbrust (1986). An Equivalence‐Theoretic Equivalent of the Axiom of Choice. Mathematical Logic Quarterly 32 (6):95-95.
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  7. Roberto Arpaia (2005). Contributions to the History of the Axiom of Foundation. Epistemologia 28 (1).
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  8. Roberto Arpaia (2005). On the Negation of the Axiom of Foundation. Epistemologia 28 (2).
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  9. 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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  10. 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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  11. 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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  12. Bernhard Banaschewski (1961). On Some Theorems Equivalent with the Axiom of Choice. Mathematical Logic Quarterly 7 (17‐18):279-282.
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  13. 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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  14. 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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  15. 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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  16. Robert Earl Brandford (1971). Cardinal Addition and the Axiom of Choice. Annals of Mathematical Logic 3 (2):111-196.
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  17. 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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  18. 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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  19. Everett L. Bull (1978). Successive Large Cardinals. Annals of Mathematical Logic 15 (2):161-191.
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  20. 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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  21. 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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  22. Paul E. Cohen (1975). A Large Power Set Axiom. Journal of Symbolic Logic 40 (1):48-54.
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  23. Omar De la Cruz, Eric J. Hall, Paul Howard, Kyriakos Keremedis & Jean E. Rubin (2008). Unions and the Axiom of Choice. Mathematical Logic Quarterly 54 (6):652-665.
    We study statements about countable and well-ordered unions and their relation to each other and to countable and well-ordered forms of the axiom of choice. Using WO as an abbreviation for “well-orderable”, here are two typical results: The assertion that every WO family of countable sets has a WO union does not imply that every countable family of WO sets has a WO union; the axiom of choice for WO families of WO sets does not imply that the countable union (...)
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  24. Jairo José Da Silva (2002). The Axioms of Set Theory. Axiomathes 13 (2):107-126.
    In this paper I argue for the view that the axioms of ZF are analytic truths of a particular concept of set. By this I mean that these axioms are true by virtue only of the meaning attached to this concept, and, moreover, can be derived from it. Although I assume that the object of ZF is a concept of set, I refrain from asserting either its independent existence, or its dependence on subjectivity. All I presuppose is that this concept (...)
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  25. James D. Davis (1972). The Inconsistency of a Certain Axiom System for Set Theory. Journal of Symbolic Logic 37 (3):538-542.
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  26. O. De la Cruz, Paul Howard & E. Hall (2002). Products of Compact Spaces and the Axiom of Choice. Mathematical Logic Quarterly 48 (4):508-516.
    We study the Tychonoff Compactness Theorem for several different definitions of a compact space.
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  27. Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis & Jean E. Rubin (2003). Metric Spaces and the Axiom of Choice. Mathematical Logic Quarterly 49 (5):455-466.
    We study conditions for a topological space to be metrizable, properties of metrizable spaces, and the role the axiom of choice plays in these matters.
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  28. Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis & Jean E. Rubin (2003). Products of Compact Spaces and the Axiom of Choice II. Mathematical Logic Quarterly 49 (1):57-71.
    This is a continuation of [2]. We study the Tychonoff Compactness Theorem for various definitions of compactness and for various types of spaces . We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.
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  29. Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis & Eleftherios Tachtsis (2005). Properties of the Real Line and Weak Forms of the Axiom of Choice. Mathematical Logic Quarterly 51 (6):598-609.
    We investigate, within the framework of Zermelo-Fraenkel set theory ZF, the interrelations between weak forms of the Axiom of Choice AC restricted to sets of reals.
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  30. J. W. Degen (1994). Some Aspects and Examples of Infinity Notions. Mathematical Logic Quarterly 40 (1):111-124.
    I wish to thank Klaus Kühnle who streamlined in [8] several of my definitions and proofs concerning the subject matter of this paper. Some ideas and results arose from discussions with Klaus Leeb. Jan Johannsen discovered some mistakes in an earlier version.
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  31. W. Degen (2002). Factors of Functions, AC and Recursive Analogues. Mathematical Logic Quarterly 48 (1):73-86.
    We investigate certain statements about factors of unary functions which have connections with weak forms of the axiom of choice. We discuss more extensively the fine structure of Howard and Rubin's Form 314 from [4]. Some of our set-theoretic results have also interesting recursive versions.
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  32. W. Degen (2001). Rigit Unary Functions and the Axiom of Choice. Mathematical Logic Quarterly 47 (2):197-204.
    We shall investigate certain statements concerning the rigidity of unary functions which have connections with forms of the axiom of choice.
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  33. Oliver Deiser (2011). An Axiomatic Theory of Well-Orderings. Review of Symbolic Logic 4 (2):186-204.
    We introduce a new simple first-order framework for theories whose objects are well-orderings (lists). A system ALT (axiomatic list theory) is presented and shown to be equiconsistent with ZFC (Zermelo Fraenkel Set Theory with the Axiom of Choice). The theory sheds new light on the power set axiom and on Gs axiom of constructibility. In list theory there are strong arguments favoring Gs axiom, while a bare analogon of the set theoretic power set axiom looks artificial. In fact, there is (...)
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  34. Christian Delhommé & Marianne Morillon (2006). Spanning Graphs and the Axiom of Choice. Reports on Mathematical Logic:165-180.
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  35. Keith J. Devlin (1973). Some Weak Versions of Large Cardinal Axioms. Annals of Mathematical Logic 5 (4):291-325.
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  36. K. Diener (2000). On Kappa-Hereditary Sets and Consequences of the Axiom of Choice. 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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  37. Karl‐Heinz Diener (1994). A Remark on Ascending Chain Conditions, the Countable Axiom of Choice and the Principle of Dependent Choices. Mathematical Logic Quarterly 40 (3):415-421.
    It is easy to prove in ZF− that a relation R satisfies the maximal condition if and only if its transitive hull R* does; equivalently: R is well-founded if and only if R* is. We will show in the following that, if the maximal condition is replaced by the chain condition, as is often the case in Algebra, the resulting statement is not provable in ZF− anymore . More precisely, we will prove that this statement is equivalent in ZF− to (...)
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  38. Karl‐Heinz Diener (1983). On Constructing Infinitary Languages Lα Β Without the Axiom of Choice. Mathematical Logic Quarterly 29 (6):357-376.
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  39. Marcel Erné (2001). Constructive Order Theory. Mathematical Logic Quarterly 47 (2):211-222.
    We introduce the notion of constructive suprema and of constructively directed sets. The Axiom of Choice turns out to be equivalent to the postulate that every supremum is constructive, but also to the hypothesis that every directed set admits a function assigning to each finite subset an upper bound. The Axiom of Multiple Choice implies a simple set-theoretical induction principle , stating that any system of sets that is closed under unions of well-ordered subsystems and contains all finite subsets of (...)
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  40. Olivier Esser (2003). On the Axiom of Extensionality in the Positive Set Theory. Mathematical Logic Quarterly 49 (1):97-100.
    This is a study of the relative interpretability of the axiom of extensionality in the positive set theory. This work has to be considered in the line of works of R. O. Gandy, D. Scott and R. Hinnion who have studied the relative interpretability of the axiom of extensionality in set theories of Zermelo and Zermelo-Fraenkel.
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  41. Harvey Friedman (2000). Does Mathematics Need New Axioms? Bulletin of Symbolic Logic 6 (4):401 - 446.
    Since about 1925, the standard formalization of mathematics has been the ZFC axiom system (Zermelo Frankel set theory with the axiom of choice), about which the audience needs to know nothing. The axiom of choice was controversial for a while, but the controversy subsided decades ago.
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  42. 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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  43. Sy D. Friedman (2005). Genericity and Large Cardinals. Journal of Mathematical Logic 5 (02):149-166.
  44. 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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  45. R. O. Gandy (1959). On the Axiom of Extensionality, Part II. Journal of Symbolic Logic 24 (4):287-300.
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  46. R. O. Gandy (1956). On the Axiom of Extensionality--Part I. Journal of Symbolic Logic 21 (1):36-48.
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  47. René Gazzari (2014). Direct Proofs of Lindenbaum Conditionals. Logica Universalis 8 (3-4):321-343.
    We discuss the problem raised by Miller to re-prove the well-known equivalences of some Lindenbaum theorems for deductive systems without an application of the Axiom of Choice. We present five special constructions of deductive systems, each of them providing some partial solutions to the mathematical problem. We conclude with a short discussion of the underlying philosophical problem of deciding, whether a given proof satisfies our demand that the Axiom of Choice is not applied.
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  48. B. Germansky (1961). The Induction Axiom and the Axiom of Choice. Mathematical Logic Quarterly 7 (11‐14):219-223.
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  49. Victoria Gitman (2011). Ramsey-Like Cardinals. Journal of Symbolic Logic 76 (2):519 - 540.
    One of the numerous characterizations of a Ramsey cardinal κ involves the existence of certain types of elementary embeddings for transitive sets of size κ satisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent with (...)
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  50. M. D. Gladstone (1968). A Single‐Axiom Impligational Calculus of Given Unsolvability. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 14 (13‐17):193-204.
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