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  1. Choice-Free Stone Duality.Nick Bezhanishvili & Wesley H. Holliday - 2020 - Journal of Symbolic Logic 85 (1):109-148.
    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 article, 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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  2. High-Order Metaphysics as High-Order Abstractions and Choice in Set Theory.Vasil Penchev - 2020 - Epistemology eJournal (Elsevier: SSRN) 13 (21):1-3.
    The link between the high-order metaphysics and abstractions, on the one hand, and choice in the foundation of set theory, on the other hand, can distinguish unambiguously the “good” principles of abstraction from the “bad” ones and thus resolve the “bad company problem” as to set theory. Thus it implies correspondingly a more precise definition of the relation between the axiom of choice and “all company” of axioms in set theory concerning directly or indirectly abstraction: the principle of abstraction, axiom (...)
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  3. Quantum Invariance.Vasil Penchev - 2020 - Epistemology eJournal (Elsevier: SSRN) 13 (22):1-6.
    Quantum invariance designates the relation of any quantum coherent state to the corresponding statistical ensemble of measured results. The adequate generalization of ‘measurement’ is discussed to involve the discrepancy, due to the fundamental Planck constant, between any quantum coherent state and its statistical representation as a statistical ensemble after measurement. A set-theory corollary is the curious invariance to the axiom of choice: Any coherent state excludes any well-ordering and thus excludes also the axiom of choice. It should be equated to (...)
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  4. The Gödel Incompleteness Theorems (1931) by the Axiom of Choice.Vasil Penchev - 2020 - Econometrics: Mathematical Methods and Programming eJournal (Elsevier: SSRN) 13 (39):1-4.
    Those incompleteness theorems mean the relation of (Peano) arithmetic and (ZFC) set theory, or philosophically, the relation of arithmetical finiteness and actual infinity. The same is managed in the framework of set theory by the axiom of choice (respectively, by the equivalent well-ordering "theorem'). One may discuss that incompleteness form the viewpoint of set theory by the axiom of choice rather than the usual viewpoint meant in the proof of theorems. The logical corollaries from that "nonstandard" viewpoint the relation of (...)
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  5. Choice, Infinity, and Negation: Both Set-Theory and Quantum-Information Viewpoints to Negation.Vasil Penchev - 2020 - Logic and Philosophy of Mathematics eJournal 12 (14):1-3.
    The concepts of choice, negation, and infinity are considered jointly. The link is the quantity of information interpreted as the quantity of choices measured in units of elementary choice: a bit is an elementary choice between two equally probable alternatives. “Negation” supposes a choice between it and confirmation. Thus quantity of information can be also interpreted as quantity of negations. The disjunctive choice between confirmation and negation as to infinity can be chosen or not in turn: This corresponds to set-theory (...)
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  6. The Axiom of Choice and the Road Paved by Sierpiński.Valérie Lynn Therrien - 2020 - Hopos: The Journal of the International Society for the History of Philosophy of Science 10 (2):504-523.
  7. 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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  8. 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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  9. 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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  10. 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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  11. 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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  12. 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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  13. Canonical Universes and Intuitions About Probabilities.Randall Dougherty & Jan Mycielski - 2006 - Dialectica 60 (4):357–368.
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  14. 4. The Axiom Of Choice And Inference To The Best Explanation.Thomas Forster - 2006 - Logique Et Analyse 49.
  15. 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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  16. 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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  17. 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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  18. 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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  19. 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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  20. On Specker's Refutation of the Axiom of Choice.Maurice Boffa - 2000 - Logique Et Analyse 43.
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  21. 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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  22. 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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  23. 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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  24. 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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  25. 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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  26. 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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  27. 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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  28. 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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  29. 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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  30. Plural Quantification and the Axiom of Choice.Stephen Pollard - 1988 - Philosophical Studies 54 (3):393 - 397.
  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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  32. Ultrapowers Without the Axiom of Choice.Mitchell Spector - 1988 - Journal of Symbolic Logic 53 (4):1208-1219.
    A new method is presented for constructing models of set theory, using a technique of forming pseudo-ultrapowers. In the presence of the axiom of choice, the traditional ultrapower construction has proven to be extremely powerful in set theory and model theory; if the axiom of choice is not assumed, the fundamental theorem of ultrapowers may fail, causing the ultrapower to lose almost all of its utility. The pseudo-ultrapower is designed so that the fundamental theorem holds even if choice fails; this (...)
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  33. 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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  34. Criteria of Identity and the Axiom of Choice.Timothy Williamson - 1986 - Journal of Philosophy 83 (7):380-394.
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  35. The Status of the Axiom of Choice in Set Theory with a Universal Set.T. E. Forster - 1985 - Journal of Symbolic Logic 50 (3):701-707.
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  36. Subgroups of a Free Group and the Axiom of Choice.Paul E. Howard - 1985 - Journal of Symbolic Logic 50 (2):458-467.
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  37. Typical Ambiguity and the Axiom of Choice.Marcel Crabbé - 1984 - Journal of Symbolic Logic 49 (4):1074-1078.
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  38. Sequential Compactness and the Axiom of Choice.Norbert Brunner - 1983 - Notre Dame Journal of Formal Logic 24 (1):89-92.
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  39. The Axiom of Choice in Topology.Norbert Brunner - 1983 - Notre Dame Journal of Formal Logic 24 (3):305-317.
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  40. On Generic Extensions Without the Axiom of Choice.G. P. Monro - 1983 - 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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  41. Lebesgue's Measure Problem and Zermelo's Axiom of Choice.Gregory H. Moore - 1983 - In Joseph Warren Dauben & Virginia Staudt Sexton (eds.), History and Philosophy of Science: Selected Papers. New York Academy of Sciences.
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  42. Equivalents of a Weak Axiom of Choice.Melven Krom - 1981 - Notre Dame Journal of Formal Logic 22 (3):283-285.
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  43. Independence Results for Class Forms of the Axiom of Choice.Paul E. Howard, Arthur L. Rubin & Jean E. Rubin - 1978 - 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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  44. Le'sniewski's Ontology Extended with the Axiom of Choice.James George Kowalski - 1977 - Notre Dame Journal of Formal Logic 18 (1):1-78.
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  45. 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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  46. A Note on the Axiom of Choice and the Continuum Hypothesis.Rolf Schock - 1977 - Notre Dame Journal of Formal Logic 18 (3):409-414.
  47. ECH, T. J.: "The Axiom of Choice". [REVIEW]John Bell - 1976 - British Journal for the Philosophy of Science 27:187.
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  48. 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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  49. An Investigation Concerning the Hilbert-Sierpi'nski Logical Form of the Axiom of Choice.Charles C. Davis - 1975 - Notre Dame Journal of Formal Logic 16 (2):145-184.
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  50. Two Forms of the Axiom of Choice for an Elementary Topos.Anna Michaelides Penk - 1975 - Journal of Symbolic Logic 40 (2):197-212.
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