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  1. 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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  2. 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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  3. 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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  4. 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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  5. Øystein Linnebo (2012). Review of P. Maddy, Defending the Axioms: On the Philosophical Foundations of Set Theory. [REVIEW] Philosophy 87 (01):133-137.
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  6. B. Lowe (2003). A Second Glance at Non-Restrictiveness. Philosophia Mathematica 11 (3):323-331.
    We give an example of a theory that strongly maximizes over ZFC and discuss possible consequences of this finding.
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  7. Donald A. Martin (2005). Gödel's Conceptual Realism. Bulletin of Symbolic Logic 11 (2):207-224.
  8. C. McLarty (2013). Penelope Maddy. Defending the Axioms: On the Philosophical Foundations of Set Theory. Oxford: Oxford University Press, 2011. ISBN 978-0-19-959618-8 (Hbk); 978-0-19-967148-9 (Pbk). Pp. X + 150. [REVIEW] Philosophia Mathematica 21 (3):385-392.
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  9. Alexander Paseau (2007). Boolos on the Justification of Set Theory. Philosophia Mathematica 15 (1):30-53.
    George Boolos has argued that the iterative conception of set justifies most, but not all, the ZFC axioms, and that a second conception of set, the Frege-von Neumann conception (FN), justifies the remaining axioms. This article challenges Boolos's claim that FN does better than the iterative conception at justifying the axioms in question. For comments on earlier versions, I am grateful to Alex Oliver, Mary Leng, Michael Potter, Øystein Linnebo, Paul Benacerraf, Peter Smith, and three journal referees.
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  10. Michael D. Resnik (1999). Review of G. Boolos, Logic, Logic, and Logic. Philosophia Mathematica 7 (3):328-335.
  11. Adam Rieger (2000). An Argument for Finsler-Aczel Set Theory. Mind 109 (434):241-253.
    Recent interest in non-well-founded set theories has been concentrated on Aczel's anti-foundation axiom AFA. I compare this axiom with some others considered by Aczel, and argue that another axiom, FAFA, is superior in that it gives the richest possible universe of sets consistent with respecting the spirit of extensionality. I illustrate how using FAFA instead of AFA might result in an improvement to Barwise and Etchemendy's treatment of the liar paradox.
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  12. Gabriel Uzquiano (2005). Review of M. Potter, Set Theory and its Philosophy: A Critical Introduction. [REVIEW] Philosophia Mathematica 13 (3):308-346.
The Axiom of Choice
  1. Alexander Abian & Wael A. Amin (1990). An Equivalent of the Axiom of Choice in Finite Models of the Powerset Axiom. Notre Dame Journal of Formal Logic 31 (3):371-374.
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  2. J. L. Bell, A Geometric Form of the Axiom of Choice.
    Consider the following well-known result from the theory of normed linear spaces ([2], p. 80, 4(b)): (g) the unit ball of the (continuous) dual of a normed linear space over the reals has an extreme point. The standard proof of (~) uses the axiom of choice (AG); thus the implication AC~(w) can be proved in set theory. In this paper we show that this implication can be reversed, so that (*) is actually eq7I2valent to the axiom of choice. From this (...)
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  3. John Bell, The Axiom of Choice in the Foundations of Mathematics.
    The principle of set theory known as the Axiom of Choice (AC) has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid’s axiom of parallels which was introduced more than two thousand years ago”1 It has been employed in countless mathematical papers, a number of monographs have been exclusively devoted to it, and it has long played a prominently role in discussions on the foundations of (...)
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  4. John Bell (2008). The Axiom of Choice and the Law of Excluded Middle in Weak Set Theories. Mathematical Logic Quarterly 54 (2):194-201.
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  5. John L. Bell (2014). Review of G. H. Moore, Zermelo's Axiom of Choice: Its Origins, Development, and Influence. [REVIEW] Philosophia Mathematica 22 (1):131-134.
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  6. John L. Bell, The Axiom of Choice. Stanford Encyclopedia of Philosophy.
    The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of (...)
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  7. Stefano Berardi, Marc Bezem & Thierry Coquand (1998). On the Computational Content of the Axiom of Choice. Journal of Symbolic Logic 63 (2):600-622.
    We present a possible computational content of the negative translation of classical analysis with the Axiom of (countable) Choice. Interestingly, this interpretation uses a refinement of the realizability semantics of the absurdity proposition, which is not interpreted as the empty type here. We also show how to compute witnesses from proofs in classical analysis of ∃-statements and how to extract algorithms from proofs of ∀∃-statements. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation.
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  8. Norbert Brunner (1983). Sequential Compactness and the Axiom of Choice. Notre Dame Journal of Formal Logic 24 (1):89-92.
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  9. Norbert Brunner (1983). The Axiom of Choice in Topology. Notre Dame Journal of Formal Logic 24 (3):305-317.
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  10. J. Richard Büchi (1953). Investigation of the Equivalence of the Axiom of Choice and Zorn's Lemma From the Viewpoint of the Hierarchy of Types. Journal of Symbolic Logic 18 (2):125-135.
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  11. J. Richard Buchi (1953). Investigation of the Equivalence of the Axiom of Choice and Zorn's Lemma From the Viewpoint of the Hierarchy of Types. Journal of Symbolic Logic 18 (2):125 - 135.
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  12. Andrea Cantini (2003). The Axiom of Choice and Combinatory Logic. Journal of Symbolic Logic 68 (4):1091-1108.
    We combine a variety of constructive methods (including forcing, realizability, asymmetric interpretation), to obtain consistency results concerning combinatory logic with extensionality and (forms of) the axiom of choice.
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  13. George E. Collins (1954). Distributivity and an Axiom of Choice. Journal of Symbolic Logic 19 (4):275-277.
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  14. Marcel Crabbé (1984). Typical Ambiguity and the Axiom of Choice. Journal of Symbolic Logic 49 (4):1074-1078.
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  15. Ch C. Davis (1975). An Investigation Concerning the Hilbert-Sierpi'nski Logical Form of the Axiom of Choice. Notre Dame Journal of Formal Logic 16 (2):145-184.
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  16. Charles C. Davis (1976). A Note on the Axiom of Choice in Leśniewski's Ontology. Notre Dame Journal of Formal Logic 17 (1):35-43.
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  17. Omar De la Cruz, Eric Hall, Paul Howard, Jean E. Rubin & Adrienne Stanley (2002). Definitions of Compactness and the Axiom of Choice. Journal of Symbolic Logic 67 (1):143-161.
    We study the relationships between definitions of compactness in topological spaces and the roll the axiom of choice plays in these relationships.
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  18. Randall Dougherty & Jan Mycielski (2006). Canonical Universes and Intuitions About Probabilities. Dialectica 60 (4):357–368.
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  19. Olivier Esser (2000). Inconsistency of the Axiom of Choice with the Positive Theory GPK+ ∞. Journal of Symbolic Logic 65 (4):1911 - 1916.
    The idea of the positive theory is to avoid the Russell's paradox by postulating an axiom scheme of comprehension for formulas without "too much" negations. In this paper, we show that the axiom of choice is inconsistent with the positive theory GPK + ∞.
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  20. T. E. Forster (1985). The Status of the Axiom of Choice in Set Theory with a Universal Set. Journal of Symbolic Logic 50 (3):701-707.
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  21. William J. Frascella (1966). The Construction of a Steiner Triple System on Sets of the Power of the Continuum Without the Axiom of Choice. Notre Dame Journal of Formal Logic 7 (2):196-202.
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  22. William J. Frascella (1965). A Generalization of Sierpiński's Theorem on Steiner Triples and the Axiom of Choice. Notre Dame Journal of Formal Logic 6 (3):163-179.
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  23. William J. Frascella (1965). Corrigendum and Addendum To: ``A Generalization of Sierpiński's Theorem on Steiner Triples and the Axiom of Choice''. Notre Dame Journal of Formal Logic 6 (4):323-324.
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  24. 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;.
  25. Lorenz Halbeisen & Saharon Shelah (2001). Relations Between Some Cardinals in the Absence of the Axiom of Choice. Bulletin of Symbolic Logic 7 (2):237-261.
    If we assume the axiom of choice, then every two cardinal numbers are comparable, In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using (...)
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  26. Jaako Hintikka (1999). Is the Axiom of Choice a Logical or Set-Theoretical Principle? Dialectica 53 (3-4):283–290.
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  27. Paul E. Howard (1992). The Axiom of Choice for Countable Collections of Countable Sets Does Not Imply the Countable Union Theorem. Notre Dame Journal of Formal Logic 33 (2):236-243.
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  28. Paul E. Howard (1985). Subgroups of a Free Group and the Axiom of Choice. Journal of Symbolic Logic 50 (2):458-467.
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  29. Paul E. Howard, Arthur L. Rubin & Jean E. Rubin (1978). Independence Results for Class Forms of the Axiom of Choice. Journal of Symbolic Logic 43 (4):673-684.
    Let NBG be von Neumann-Bernays-Gödel set theory without the axiom of choice and let NBGA be the modification which allows atoms. In this paper we consider some of the well-known class or global forms of the wellordering theorem, the axiom of choice, and maximal principles which are known to be equivalent in NBG and show they are not equivalent in NBGA.
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  30. Paul E. Howard & Mary Yorke (1987). Maximal $P$-Subgroups and the Axiom of Choice. Notre Dame Journal of Formal Logic 28 (2):276-283.
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  31. Paul Howard & Jean E. Rubin (1995). The Axiom of Choice for Well-Ordered Families and for Families of Well- Orderable Sets. Journal of Symbolic Logic 60 (4):1115-1117.
    We show that it is not possible to construct a Fraenkel-Mostowski model in which the axiom of choice for well-ordered families of sets and the axiom of choice for sets are both true, but the axiom of choice is false.
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  32. James Kowalski (1977). Le'sniewski's Ontology Extended with the Axiom of Choice. Notre Dame Journal of Formal Logic 18 (1):1-78.
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  33. Melven Krom (1981). Equivalents of a Weak Axiom of Choice. Notre Dame Journal of Formal Logic 22 (3):283-285.
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  34. Gabriele Lolli (1977). On Ramsey's Theorem and the Axiom of Choice. Notre Dame Journal of Formal Logic 18 (4):599-601.
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  35. Elliott Mendelson (1956). The Independence of a Weak Axiom of Choice. Journal of Symbolic Logic 21 (4):350-366.
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  36. David W. Miller (2007). Some Restricted Lindenbaum Theorems Equivalent to the Axiom of Choice. Logica Universalis 1 (1):183-199.
    . Dzik [2] gives a direct proof of the axiom of choice from the generalized Lindenbaum extension theorem LET. The converse is part of every decent logical education. Inspection of Dzik’s proof shows that its premise let attributes a very special version of the Lindenbaum extension property to a very special class of deductive systems, here called Dzik systems. The problem therefore arises of giving a direct proof, not using the axiom of choice, of the conditional . A partial solution (...)
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  37. G. Mints (1999). Cut-Elimination for Simple Type Theory with an Axiom of Choice. Journal of Symbolic Logic 64 (2):479-485.
    We present a cut-elimination proof for simple type theory with an axiom of choice formulated in the language with an epsilon-symbol. The proof is modeled after Takahashi's proof of cut-elimination for simple type theory with extensionality. The same proof works when types are restricted, for example for second-order classical logic with an axiom of choice.
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  38. G. P. Monro (1983). On Generic Extensions Without the Axiom of Choice. Journal of Symbolic Logic 48 (1):39-52.
    Let ZF denote Zermelo-Fraenkel set theory (without the axiom of choice), and let $M$ be a countable transitive model of ZF. The method of forcing extends $M$ to another model $M\lbrack G\rbrack$ of ZF (a "generic extension"). If the axiom of choice holds in $M$ it also holds in $M\lbrack G\rbrack$, that is, the axiom of choice is preserved by generic extensions. We show that this is not true for many weak forms of the axiom of choice, and we derive (...)
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