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  1. added 2019-08-11
    Neuer Beweis für die Möglichkeit einer Wohlordnung.Ernst Zermelo - 1907 - Mathematische Annalen 65 (1):107-128.
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  2. added 2019-07-03
    Choice-Free Stone Duality.Nick Bezhanishvili & Wesley H. Holliday - forthcoming - Journal of Symbolic Logic.
    The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this paper, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean (...)
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  3. added 2019-06-05
    Gregory H. Moore. Zermelo’s Axiom of Choice: Its Origins, Development, and Influence. Mineola, N.Y.: Dover Publications, 2013. ISBN 978-0-48648841-7 . Pp. 448: Critical Studies/Book Reviews. [REVIEW]John L. Bell - 2014 - Philosophia Mathematica 22 (1):131-134.
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  4. added 2019-06-05
    Inconsistency of the Axiom of Choice with the Positive Theory $GPK^+ \Infty$.Olivier Esser - 2000 - 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 $GPK^+ \infty$.
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  5. added 2019-06-05
    Corrigendum and Addendum to My Paper "A Generalization of Sierpinski's Theorem on Steiner Triples and the Axiom of Choice".William J. Frascella - 1965 - Notre Dame Journal of Formal Logic 6:323.
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  6. added 2016-09-09
    Russell’s Method of Analysis and the Axioms of Mathematics.Lydia Patton - 2017 - In Sandra Lapointe Christopher Pincock (ed.), Innovations in the History of Analytical Philosophy. London: Palgrave-Macmillan. pp. 105-126.
    In the early 1900s, Russell began to recognize that he, and many other mathematicians, had been using assertions like the Axiom of Choice implicitly, and without explicitly proving them. In working with the Axioms of Choice, Infinity, and Reducibility, and his and Whitehead’s Multiplicative Axiom, Russell came to take the position that some axioms are necessary to recovering certain results of mathematics, but may not be proven to be true absolutely. The essay traces historical roots of, and motivations for, Russell’s (...)
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  7. added 2015-04-21
    On Specker's Refutation of the Axiom of Choice.Maurice Boffa - 2000 - Logique Et Analyse 43.
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  8. added 2015-04-15
    ECH, T. J.: "The Axiom of Choice". [REVIEW]John Bell - 1976 - British Journal for the Philosophy of Science 27:187.
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  9. added 2015-03-17
    M L.John L. Bell - unknown
    A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that beefing up WST with moderate extensionality principles or quotient sets enables the derivation to go through.
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  10. added 2015-03-17
    Some Consequences of the Axiom of Power-Set.Alexander Abian & Samuel Lamacchia - 1965 - Journal of Symbolic Logic 30 (3):293-294.
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  11. added 2014-04-02
    The Axiom of Choice for Well-Ordered Families and for Families of Well- Orderable Sets.Paul Howard & Jean E. Rubin - 1995 - 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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  12. added 2014-03-28
    Is the Axiom of Choice a Logical or Set-Theoretical Principle?Jaako Hintikka - 1999 - Dialectica 53 (3-4):283–290.
    A generalization of the axioms of choice says that all the Skolem functions of a true first‐order sentence exist. This generalization can be implemented on the first‐order level by generalizing the rule of existential instantiation into a rule of functional instantiation. If this generalization is carried out in first‐order axiomatic set theory , it is seen that in any model of FAST, there are sentences S which are true but whose Skolem functions do not exist. Since this existence is what (...)
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  13. added 2014-03-25
    Relations Between Some Cardinals in the Absence of the Axiom of Choice.Lorenz Halbeisen & Saharon Shelah - 2001 - 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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  14. added 2014-03-24
    Definitions of Compactness and the Axiom of Choice.Omar De la Cruz, Eric Hall, Paul Howard, Jean E. Rubin & Adrienne Stanley - 2002 - 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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  15. added 2014-03-22
    The Axiom of Choice and Combinatory Logic.Andrea Cantini - 2003 - 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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  16. added 2014-03-18
    Canonical Universes and Intuitions About Probabilities.Randall Dougherty & Jan Mycielski - 2006 - Dialectica 60 (4):357–368.
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  17. added 2014-03-12
    The Axiom of Choice and the Law of Excluded Middle in Weak Set Theories.John L. Bell - 2008 - Mathematical Logic Quarterly 54 (2):194-201.
    A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that bee.ng up WST with moderate extensionality principles or quotient sets enables the derivation to go through.
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  18. added 2014-03-11
    Countable Choice as a Questionable Uniformity Principle.Peter M. Schuster - 2004 - Philosophia Mathematica 12 (2):106-134.
    The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.
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  19. added 2014-03-06
    Notions of Compactness for Special Subsets of ℝ I and Some Weak Forms of the Axiom of Choice.Marianne Morillon - 2010 - Journal of Symbolic Logic 75 (1):255-268.
    We work in set-theory without choice ZF. A set is Countable if it is finite or equipotent with ${\Bbb N}$ . Given a closed subset F of [0, 1] I which is a bounded subset of $\ell ^{1}(I)$ (resp. such that $F\subseteq c_{0}(I)$ ), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice AC N ) implies that F is compact. This enhances previous results where AC N (resp. the axiom of Dependent (...)
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  20. added 2011-05-31
    A Geometric Form of the Axiom of Choice.J. L. Bell - unknown
    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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  21. added 2011-05-31
    The Axiom of Choice for Countable Collections of Countable Sets Does Not Imply the Countable Union Theorem.Paul E. Howard - 1992 - Notre Dame Journal of Formal Logic 33 (2):236-243.
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  22. added 2011-05-31
    A Note on Some Weak Forms of the Axiom of Choice.Gary P. Shannon - 1991 - Notre Dame Journal of Formal Logic 33 (1):144-147.
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  23. added 2011-05-31
    An Equivalent of the Axiom of Choice in Finite Models of the Powerset Axiom.Alexander Abian & Wael A. Amin - 1990 - Notre Dame Journal of Formal Logic 31 (3):371-374.
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  24. added 2011-05-31
    Equivalent Versions of a Weak Form of the Axiom of Choice.Gary P. Shannon - 1988 - Notre Dame Journal of Formal Logic 29 (4):569-573.
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  25. added 2011-05-31
    Maximal $P$-Subgroups and the Axiom of Choice.Paul E. Howard & Mary Yorke - 1987 - Notre Dame Journal of Formal Logic 28 (2):276-283.
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  26. added 2011-05-31
    Sequential Compactness and the Axiom of Choice.Norbert Brunner - 1983 - Notre Dame Journal of Formal Logic 24 (1):89-92.
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  27. added 2011-05-31
    The Axiom of Choice in Topology.Norbert Brunner - 1983 - Notre Dame Journal of Formal Logic 24 (3):305-317.
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  28. added 2011-05-31
    Equivalents of a Weak Axiom of Choice.Melven Krom - 1981 - Notre Dame Journal of Formal Logic 22 (3):283-285.
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  29. added 2011-05-31
    A Note on the Axiom of Choice and the Continuum Hypothesis.Rolf Schock - 1977 - Notre Dame Journal of Formal Logic 18 (3):409-414.
  30. added 2011-05-31
    On Ramsey's Theorem and the Axiom of Choice.Gabriele Lolli - 1977 - Notre Dame Journal of Formal Logic 18 (4):599-601.
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  31. added 2011-05-31
    Le'sniewski's Ontology Extended with the Axiom of Choice.James Kowalski - 1977 - Notre Dame Journal of Formal Logic 18 (1):1-78.
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  32. added 2011-05-31
    A Note on the Axiom of Choice in Leśniewski's Ontology.Charles C. Davis - 1976 - Notre Dame Journal of Formal Logic 17 (1):35-43.
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  33. added 2011-05-31
    An Investigation Concerning the Hilbert-Sierpi'nski Logical Form of the Axiom of Choice.Ch C. Davis - 1975 - Notre Dame Journal of Formal Logic 16 (2):145-184.
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  34. added 2011-05-31
    The Axiom of Choice.Thomas Jech - 1973 - North-Holland.
    Provability, Computability and Reflection.
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  35. added 2011-05-31
    A Theorem for Deriving Consequences of the Axiom of Choice.Francis J. Tytus - 1967 - Notre Dame Journal of Formal Logic 8 (4):291-296.
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  36. added 2011-05-31
    The Construction of a Steiner Triple System on Sets of the Power of the Continuum Without the Axiom of Choice.William J. Frascella - 1966 - Notre Dame Journal of Formal Logic 7 (2):196-202.
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  37. added 2011-05-31
    A Generalization of Sierpiński's Theorem on Steiner Triples and the Axiom of Choice.William J. Frascella - 1965 - Notre Dame Journal of Formal Logic 6 (3):163-179.
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  38. added 2011-05-31
    A Theorem of Sierpiński on Triads and the Axiom of Choice.Bolesław Sobociński - 1964 - Notre Dame Journal of Formal Logic 5 (1):51-58.
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  39. added 2011-05-31
    A Set-Theoretical Formula Equivalent to the Axiom of Choice.Bolesław Sobociński - 1962 - Notre Dame Journal of Formal Logic 3 (3):167-169.
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  40. added 2011-05-31
    Certain Formulas Equivalent to the Axiom of Choice.Bolesław Sobociński - 1961 - Notre Dame Journal of Formal Logic 2 (4):229-235.
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  41. added 2011-05-31
    A Simple Formula Equivalent to the Axiom of Choice.Bolesław Sobociński - 1960 - Notre Dame Journal of Formal Logic 1 (3):115-117.
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  42. added 2011-05-31
    A Note Concerning the Axiom of Choice.Bolesław Sobociński - 1960 - Notre Dame Journal of Formal Logic 1 (3):122-122.
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  43. added 2011-02-01
    Set Theory and its Philosophy: A Critical Introduction.Michael Potter - 2004 - Oxford University Press.
    Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set (...)
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  44. added 2010-05-13
    The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory.Kurt Gödel - 1940 - Princeton University Press;.
  45. added 2008-12-31
    The Axiom of Choice in the Foundations of Mathematics.John Bell - manuscript
    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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  46. added 2008-12-31
    The Axiom of Choice.John L. Bell - 2008 - 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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  47. added 2008-12-31
    Some Restricted Lindenbaum Theorems Equivalent to the Axiom of Choice.David W. Miller - 2007 - 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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  48. added 2008-12-31
    Cut-Elimination for Simple Type Theory with an Axiom of Choice.G. Mints - 1999 - 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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  49. added 2008-12-31
    On the Computational Content of the Axiom of Choice.Stefano Berardi, Marc Bezem & Thierry Coquand - 1998 - 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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  50. added 2008-12-31
    Independence, Randomness and the Axiom of Choice.Michiel van Lambalgen - 1992 - Journal of Symbolic Logic 57 (4):1274-1304.
    We investigate various ways of introducing axioms for randomness in set theory. The results show that these axioms, when added to ZF, imply the failure of AC. But the axiom of extensionality plays an essential role in the derivation, and a deeper analysis may ultimately show that randomness is incompatible with extensionality.
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