Results for 'Axiom of multiple choice'

988 found
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  1.  50
    Paracompactness of Metric Spaces and the Axiom of Multiple Choice.Paul Howard, K. Keremedis & J. E. Rubin - 2000 - Mathematical Logic Quarterly 46 (2):219-232.
    The axiom of multiple choice implies that metric spaces are paracompact but the reverse implication cannot be proved in set theory without the axiom of choice.
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  2.  62
    The axiom of multiple choice and models for constructive set theory.Benno van den Berg & Ieke Moerdijk - 2014 - Journal of Mathematical Logic 14 (1):1450005.
    We propose an extension of Aczel's constructive set theory CZF by an axiom for inductive types and a choice principle, and show that this extension has the following properties: it is interpretable in Martin-Löf's type theory. In addition, it is strong enough to prove the Set Compactness theorem and the results in formal topology which make use of this theorem. Moreover, it is stable under the standard constructions from algebraic set theory, namely exact completion, realizability models, forcing as (...)
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  3.  26
    Lévy A.. Axioms of multiple choice. Fundamenta mathematicae, vol. 50 no. 5 , pp. 475–483.J. R. Shoenfield - 1965 - Journal of Symbolic Logic 30 (2):252-252.
  4. Review: A. Levy, Axioms of Multiple Choice[REVIEW]J. R. Shoenfield - 1965 - Journal of Symbolic Logic 30 (2):252-252.
  5.  68
    The Axiom of Choice in Quantum Theory.Norbert Brunner, Karl Svozil & Matthias Baaz - 1996 - 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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  6.  29
    The Vector Space Kinna-Wagner Principle is Equivalent to the Axiom of Choice.Kyriakos Keremedis - 2001 - Mathematical Logic Quarterly 47 (2):205-210.
    We show that the axiom of choice AC is equivalent to the Vector Space Kinna-Wagner Principle, i.e., the assertion: “For every family [MATHEMATICAL SCRIPT CAPITAL V]= {Vi : i ∈ k} of non trivial vector spaces there is a family ℱ = {Fi : i ∈ k} such that for each i ∈ k, Fiis a non empty independent subset of Vi”. We also show that the statement “every vector space over ℚ has a basis” implies that every (...)
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  7.  15
    Multiple choices imply the ingleton and krein–milman axioms.Marianne Morillon - 2020 - Journal of Symbolic Logic 85 (1):439-455.
    In set theory without the Axiom of Choice, we consider Ingleton’s axiom which is the ultrametric counterpart of the Hahn–Banach axiom. We show that in ZFA, i.e., in the set theory without the Axiom of Choice weakened to allow “atoms,” Ingleton’s axiom does not imply the Axiom of Choice. We also prove that in ZFA, the “multiple choiceaxiom implies the Krein–Milman axiom. We deduce that, in ZFA, (...)
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  8.  12
    If vector spaces are projective modules then multiple choice holds.Paul Howard - 2005 - Mathematical Logic Quarterly 51 (2):187.
    We show that the assertion that every vector space is a projective module implies the axiom of multiple choice and that the reverse implication does not hold in set theory weakened to permit the existence of atoms.
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  9.  60
    Products of compact spaces and the axiom of choice II.Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis & Jean E. Rubin - 2003 - 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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  10.  60
    Disasters in topology without the axiom of choice.Kyriakos Keremedis - 2001 - Archive for Mathematical Logic 40 (8):569-580.
    We show that some well known theorems in topology may not be true without the axiom of choice.
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  11.  38
    Compact Metric Spaces and Weak Forms of the Axiom of Choice.E. Tachtsis & K. Keremedis - 2001 - Mathematical Logic Quarterly 47 (1):117-128.
    It is shown that for compact metric spaces the following statements are pairwise equivalent: “X is Loeb”, “X is separable”, “X has a we ordered dense subset”, “X is second countable”, and “X has a dense set G = ∪{Gn : n ∈ ω}, ∣Gn∣ < ω, with limn→∞ diam = 0”. Further, it is shown that the statement: “Compact metric spaces are weakly Loeb” is not provable in ZF0 , the Zermelo-Fraenkel set theory without the axiom of regularity, (...)
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  12.  11
    Bleicher M. N.. Multiple choice axioms and axioms of choice for finite sets. Fundamenta mathematicae, vol. 57 , pp. 247–252. [REVIEW]J. D. Halpern - 1967 - Journal of Symbolic Logic 32 (2):273-273.
  13.  42
    Filters, Antichains and Towers in Topological Spaces and the Axiom of Choice.Kyriakos Keremedis - 1998 - Mathematical Logic Quarterly 44 (3):359-366.
    We find some characterizations of the Axiom of Choice in terms of certain families of open sets in T1 spaces.
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  14.  14
    Review: M. N. Bleicher, Multiple Choice Axioms and Axioms of Choice for Finite Sets. [REVIEW]J. D. Halpern - 1967 - Journal of Symbolic Logic 32 (2):273-273.
  15. Russell’s method of analysis and the axioms of mathematics.Lydia Patton - 2017 - In Sandra Lapointe & Christopher Pincock (eds.), Innovations in the History of Analytical Philosophy. London, United Kingdom: 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, (...)
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  16.  9
    Constructive Order Theory.Marcel Erné - 2001 - 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 (...)
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  17.  14
    Models of $${{\textsf{ZFA}}}$$ in which every linearly ordered set can be well ordered.Paul Howard & Eleftherios Tachtsis - 2023 - Archive for Mathematical Logic 62 (7):1131-1157.
    We provide a general criterion for Fraenkel–Mostowski models of $${\textsf{ZFA}}$$ (i.e. Zermelo–Fraenkel set theory weakened to permit the existence of atoms) which implies “every linearly ordered set can be well ordered” ( $${\textsf{LW}}$$ ), and look at six models for $${\textsf{ZFA}}$$ which satisfy this criterion (and thus $${\textsf{LW}}$$ is true in these models) and “every Dedekind finite set is finite” ( $${\textsf{DF}}={\textsf{F}}$$ ) is true, and also consider various forms of choice for well-ordered families of well orderable sets in (...)
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  18.  35
    Some consequences of Rado’s selection lemma.Marianne Morillon - 2012 - Archive for Mathematical Logic 51 (7-8):739-749.
    We prove in set theory without the Axiom of Choice, that Rado’s selection lemma (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{RL}}$$\end{document}) implies the Hahn-Banach axiom. We also prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{RL}}$$\end{document} is equivalent to several consequences of the Tychonov theorem for compact Hausdorff spaces: in particular, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{RL}}$$\end{document} implies that every filter on a well orderable set is included (...)
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  19.  57
    Why humans are (sometimes) less rational than other animals: Cognitive complexity and the axioms of rational choice.Keith E. Stanovich - 2013 - Thinking and Reasoning 19 (1):1 - 26.
    (2013). Why humans are (sometimes) less rational than other animals: Cognitive complexity and the axioms of rational choice. Thinking & Reasoning: Vol. 19, No. 1, pp. 1-26. doi: 10.1080/13546783.2012.713178.
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  20.  11
    The axiom of choice in metric measure spaces and maximal $$\delta $$-separated sets.Michał Dybowski & Przemysław Górka - 2023 - Archive for Mathematical Logic 62 (5):735-749.
    We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and finite implies separability of the space. In this way a negative answer to an open problem formulated in Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal $$\delta $$ δ -separated sets in metric and pseudometric spaces from the (...)
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  21. Axioms for deliberative stit.Ming Xu - 1998 - Journal of Philosophical Logic 27 (5):505-552.
    Based on a notion of "companions to stit formulas" applied in other papers dealing with astit logics, we introduce "choice formulas" and "nested choice formulas" to prove the completeness theorems for dstit logics in a language with the dstit operator as the only non-truth-functional operator. The main logic discussed in this paper is the basic logic of dstit with multiple agents, other logics discussed include the basic logic of dstit with a single agent and some logics of (...)
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  22.  8
    A gradient theory of multiple-choice learning.John Oliver Cook - 1953 - Psychological Review 60 (1):15-22.
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  23. 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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  24.  39
    The axiom of choice holds iff maximal closed filters exist.Horst Herrlich - 2003 - Mathematical Logic Quarterly 49 (3):323.
    It is shown that in ZF set theory the axiom of choice holds iff every non empty topological space has a maximal closed filter.
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  25.  16
    The difficulty of multiple choice test item alternatives.P. Horst - 1932 - Journal of Experimental Psychology 15 (4):469.
  26.  26
    On uniformly continuous functions between pseudometric spaces and the Axiom of Countable Choice.Samuel G. da Silva - 2019 - Archive for Mathematical Logic 58 (3-4):353-358.
    In this note we show that the Axiom of Countable Choice is equivalent to two statements from the theory of pseudometric spaces: the first of them is a well-known characterization of uniform continuity for functions between metric spaces, and the second declares that sequentially compact pseudometric spaces are \—meaning that all real valued, continuous functions defined on these spaces are necessarily uniformly continuous.
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  27. God's problem of multiple choice.Lloyd Strickland - 2006 - Religious Studies 42 (2):141-157.
    A question that has been largely overlooked by philosophers of religion is how God would be able to effect a rational choice between two worlds of unsurpassable goodness. To answer this question, I draw a parallel with the paradigm cases of indifferent choice, including Buridan's ass, and argue that such cases can be satisfactorily resolved provided that the protagonists employ what Otto Neurath calls an ‘auxiliary motive’. I supply rational grounds for the employment of such a motive, and (...)
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  28.  14
    The Strength of an Axiom of Finite Choice for Branches in Trees.G. O. H. Jun Le - 2023 - Journal of Symbolic Logic 88 (4):1367-1386.
    In their logical analysis of theorems about disjoint rays in graphs, Barnes, Shore, and the author (hereafter BGS) introduced a weak choice scheme in second-order arithmetic, called the $\Sigma ^1_1$ axiom of finite choice (hereafter finite choice). This is a special case of the $\Sigma ^1_1$ axiom of choice ( $\Sigma ^1_1\text {-}\mathsf {AC}_0$ ) introduced by Kreisel. BGS showed that $\Sigma ^1_1\text {-}\mathsf {AC}_0$ suffices for proving many of the aforementioned theorems in graph (...)
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  29. 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 (...)
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  30. The Axiom of Choice is False Intuitionistically (in Most Contexts).Charles Mccarty, Stewart Shapiro & Ansten Klev - 2023 - Bulletin of Symbolic Logic 29 (1):71-96.
    There seems to be a view that intuitionists not only take the Axiom of Choice (AC) to be true, but also believe it a consequence of their fundamental posits. Widespread or not, this view is largely mistaken. This article offers a brief, yet comprehensive, overview of the status of AC in various intuitionistic and constructivist systems. The survey makes it clear that the Axiom of Choice fails to be a theorem in most contexts and is even (...)
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  31. Virtue theory of mathematical practices: an introduction.Andrew Aberdein, Colin Jakob Rittberg & Fenner Stanley Tanswell - 2021 - Synthese 199 (3-4):10167-10180.
    Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and fragmentary at best. But in the last few years this has begun to change. As virtue theory has grown ever more influential, not just in ethics where virtues may seem most at home, but particularly in epistemology and the philosophy of science, some philosophers have sought to push virtues out into unexpected areas, including mathematics and its philosophy. But there are some mathematicians already there, ready to (...)
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  32.  49
    The Axiom of Choice in Second‐Order Predicate Logic.Christine Gaßner - 1994 - Mathematical Logic Quarterly 40 (4):533-546.
    The present article deals with the power of the axiom of choice within the second-order predicate logic. We investigate the relationship between several variants of AC and some other statements, known as equivalent to AC within the set theory of Zermelo and Fraenkel with atoms, in Henkin models of the one-sorted second-order predicate logic with identity without operation variables. The construction of models follows the ideas of Fraenkel and Mostowski. It is e. g. shown that the well-ordering theorem (...)
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  33. 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 (...)
     
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  34.  11
    The Axiom of Choice as Interaction Brief Remarks on the Principle of Dependent Choices in a Dialogical Setting.Shahid Rahman - 2018 - In Hassan Tahiri (ed.), The Philosophers and Mathematics: Festschrift for Roshdi Rashed. Cham: Springer Verlag. pp. 201-248.
    The work of Roshdi Rashed has set a landmark in many senses, but perhaps the most striking one is his inexhaustible thrive to open new paths for the study of conceptual links between science and philosophy deeply rooted in the interaction of historic with systematic perspectives. In the present talk I will focus on how a framework that has its source in philosophy of logic, interacts with some new results on the foundations of mathematics. More precisely, the main objective of (...)
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  35.  81
    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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  36. The axiom of determinancy implies dependent choices in l(r).Alexander S. Kechris - 1984 - Journal of Symbolic Logic 49 (1):161 - 173.
    We prove the following Main Theorem: $ZF + AD + V = L(R) \Rightarrow DC$ . As a corollary we have that $\operatorname{Con}(ZF + AD) \Rightarrow \operatorname{Con}(ZF + AD + DC)$ . Combined with the result of Woodin that $\operatorname{Con}(ZF + AD) \Rightarrow \operatorname{Con}(ZF + AD + \neg AC^\omega)$ it follows that DC (as well as AC ω ) is independent relative to ZF + AD. It is finally shown (jointly with H. Woodin) that ZF + AD + ¬ DC (...)
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  37.  54
    Weak Forms of the Axiom of Choice and the Generalized Continuum Hypothesis.Arthur L. Rubin & Jean E. Rubin - 1993 - Mathematical Logic Quarterly 39 (1):7-22.
    In this paper we study some statements similar to the Partition Principle and the Trichotomy. We prove some relationships between these statements, the Axiom of Choice, and the Generalized Continuum Hypothesis. We also prove some independence results. MSC: 03E25, 03E50, 04A25, 04A50.
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  38.  10
    The axiom of determinacy implies dependent choice in mice.Sandra Müller - 2019 - Mathematical Logic Quarterly 65 (3):370-375.
    We show that the Axiom of Dependent Choice,, holds in countably iterable, passive premice constructed over their reals which satisfy the Axiom of Determinacy,, in a background universe. This generalizes an argument of Kechris for using Steel's analysis of scales in mice. In particular, we show that for any and any countable set of reals A so that and, we have that.
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  39.  10
    Imperfect models, imperfect conclusions: An exploratory study of multiple-choice tests and historical knowledge.Gabriel A. Reich - 2013 - Journal of Social Studies Research 37 (1):3-16.
    This article explores the extent to which multiple-choice history/social studies exams measure student knowledge of social studies content. This article presents descriptive statistics that quantify the findings from a qualitative study. Data for this study were collected from 13 tenth-grade world history students in an urban classroom in New York State. Each participant answered 15 multiple-choice questions that had appeared on previous versions of the Global History and Geography Regents exam, the high-stakes exam they would have (...)
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  40. The Axiom of choice in Quine's New Foundations for Mathematical Logic.Ernst P. Specker - 1954 - Journal of Symbolic Logic 19 (2):127-128.
  41.  25
    Shadows of the axiom of choice in the universe $$L$$.Jan Mycielski & Grzegorz Tomkowicz - 2018 - Archive for Mathematical Logic 57 (5-6):607-616.
    We show that several theorems about Polish spaces, which depend on the axiom of choice ), have interesting corollaries that are theorems of the theory \, where \ is the axiom of dependent choices. Surprisingly it is natural to use the full \ to prove the existence of these proofs; in fact we do not even know the proofs in \. Let \ denote the axiom of determinacy. We show also, in the theory \\), a theorem (...)
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  42.  95
    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 (...)
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  43.  21
    Axiom of choice and excluded middle in categorical logic.Steven Awodey - 1995 - Bulletin of Symbolic Logic 1:344.
  44.  15
    The Axiom of Choice and the Partition Principle from Dialectica Categories.Samuel G. Da Silva - forthcoming - Logic Journal of the IGPL.
    The method of morphisms is a well-known application of Dialectica categories to set theory. In a previous work, Valeria de Paiva and the author have asked how much of the Axiom of Choice is needed in order to carry out the referred applications of such method. In this paper, we show that, when considered in their full generality, those applications of Dialectica categories give rise to equivalents of either the Axiom of Choice or Partition Principle —which (...)
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  45.  46
    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.
    From 1908 to 1916, articles supporting the axiom of choice were scant. The situation changed in 1916, when Wacław Sierpiński published a series of articles reviving the debate. The posterity of the axiom of choice as we know it would be unimaginable without Sierpiński’s efforts.
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  46.  47
    Metric spaces and the axiom of choice.Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis & Jean E. Rubin - 2003 - 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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  47.  10
    The Axiom of Choice and the Class of Hyperarithmetic Functions.G. Kreisel - 1970 - Journal of Symbolic Logic 35 (2):333-334.
  48.  29
    Determinate logic and the Axiom of Choice.J. P. Aguilera - 2020 - Annals of Pure and Applied Logic 171 (2):102745.
    Takeuti introduced an infinitary proof system for determinate logic and showed that for transitive models of Zermelo-Fraenkel set theory with the Axiom of Dependent Choice that contain all reals, the cut-elimination theorem is equivalent to the Axiom of Determinacy, and in particular contradicts the Axiom of Choice. We consider variants of Takeuti's theorem without assuming the failure of the Axiom of Choice. For instance, we show that if one removes atomic formulae of infinite (...)
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  49.  38
    The Hahn-Banach Property and the Axiom of Choice.Juliette Dodu & Marianne Morillon - 1999 - Mathematical Logic Quarterly 45 (3):299-314.
    We work in set theory ZF without axiom of choice. Though the Hahn-Banach theorem cannot be proved in ZF, we prove that every Gateaux-differentiable uniformly convex Banach space E satisfies the following continuous Hahn-Banach property: if p is a continuous sublinear functional on E, if F is a subspace of E, and if f: F → ℝ is a linear functional such that f ≤ p|F then there exists a linear functional g : E → ℝ such that (...)
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  50.  70
    Unions and the axiom of choice.Omar De la Cruz, Eric J. Hall, Paul Howard, Kyriakos Keremedis & Jean E. Rubin - 2008 - 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 (...)
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