Results for 'Axiom of choice'

1000+ found
Order:
See also
  1. 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 (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   10 citations  
  2. 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 (...)
     
    Export citation  
     
    Bookmark   1 citation  
  3. 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.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  4.  10
    Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory.Kurt Gödel - 1940 - Princeton, NJ, USA: Princeton University Press.
  5.  32
    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 (...)
    Direct download  
     
    Export citation  
     
    Bookmark  
  6. The Axiom of Choice Vol. 22.John L. Bell - 2009 - College Publications.
     
    Export citation  
     
    Bookmark   1 citation  
  7.  49
    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.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  8.  15
    Axiom of Choice and Excluded Middle in Categorical Logic.Steven Awodey - 1995 - Bulletin of Symbolic Logic 1:344.
  9.  29
    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.
    Direct download  
     
    Export citation  
     
    Bookmark   1 citation  
  10.  6
    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 (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  11.  51
    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.
    Direct download (9 more)  
     
    Export citation  
     
    Bookmark  
  12. The Axiom of Choice in Quine's New Foundations for Mathematical Logic.Ernst P. Specker - 1954 - Journal of Symbolic Logic 19 (2):127-128.
  13.  58
    The Axiom of Choice is False Intuitionistically.Charles Mccarty, Stewart Shapiro & Ansten Klev - forthcoming - Bulletin of Symbolic Logic:1-26.
  14.  94
    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.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark  
  15.  23
    ZF and the Axiom of Choice in Some Paraconsistent Set Theories.Thierry Libert - 2003 - Logic and Logical Philosophy 11:91-114.
    In this paper, we present set theories based upon the paraconsistent logic Pac. We describe two different techniques to construct models of such set theories. The first of these is an adaptation of one used to construct classical models of positive comprehension. The properties of the models obtained in that way give rise to a natural paraconsistent set theory which is presented here. The status of the axiom of choice in that theory is also discussed. The second leads (...)
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  16.  7
    The Axiom of Choice and the Class of Hyperarithmetic Functions.G. Kreisel - 1970 - Journal of Symbolic Logic 35 (2):333-334.
  17.  55
    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 (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  18.  2
    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. 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 (...)
    No categories
    Direct download  
     
    Export citation  
     
    Bookmark  
  19.  16
    Axiom of Choice for Finite Sets.Andrzej Mostowski - 1948 - Journal of Symbolic Logic 13 (1):45-46.
  20.  21
    About the Axiom of Choice.Thomas J. Jech - 1977 - In Jon Barwise & H. Jerome Keisler (eds.), Handbook of Mathematical Logic. North-Holland Pub. Co.. pp. 90--345.
  21. Assertion Proof and the Axiom of Choice.David DeVidi - 2006 - In ¸ Itedevidikenyon2006. Springer Verlag.
     
    Export citation  
     
    Bookmark   3 citations  
  22.  32
    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.
    Direct download  
     
    Export citation  
     
    Bookmark   1 citation  
  23.  19
    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.
  24.  23
    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.
  25.  40
    The Axiom of Choice in Topology.Norbert Brunner - 1983 - Notre Dame Journal of Formal Logic 24 (3):305-317.
  26.  45
    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.
    Direct download  
     
    Export citation  
     
    Bookmark   2 citations  
  27. ECH, T. J.: "The Axiom of Choice". [REVIEW]John Bell - 1976 - British Journal for the Philosophy of Science 27:187.
     
    Export citation  
     
    Bookmark  
  28. Antinomicity and the Axiom of Choice.Florencio G. Asenjo - 1996 - Logic and Logical Philosophy 4:53-95.
  29.  8
    Review: G. Kreisel, The Axiom of Choice and the Class of Hyperarithmetic Functions. [REVIEW]Yiannis N. Moschovakis - 1970 - Journal of Symbolic Logic 35 (2):333-334.
  30. From the Axiom of Choice to Choice Sequences.H. Jervell - 1996 - Nordic Journal of Philosophical Logic 1 (1):95-98.
  31.  14
    The Failure of the Axiom of Choice Implies Unrest in the Theory of Lindelöf Metric Spaces.Kyriakos Keremedis - 2003 - Mathematical Logic Quarterly 49 (2):179-186.
    In the realm of metric spaces the role of choice principles is investigated.
    Direct download  
     
    Export citation  
     
    Bookmark   2 citations  
  32. 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 $GPK^+ \infty$.
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  33.  26
    Zermelo's Axiom of Choice. Its Origins, Development, and Influence.Gregory H. Moore - 1984 - Journal of Symbolic Logic 49 (2):659-660.
  34. 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.
    Direct download (10 more)  
     
    Export citation  
     
    Bookmark   8 citations  
  35.  11
    Infinitesimal Analysis Without the Axiom of Choice.Karel Hrbacek & Mikhail G. Katz - 2021 - Annals of Pure and Applied Logic 172 (6):102959.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  36.  62
    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 (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  37.  19
    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 (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  38.  8
    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 (...)
    No categories
    Direct download  
     
    Export citation  
     
    Bookmark   1 citation  
  39. 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 (...)
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  40.  18
    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 (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  41. 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 (...)
     
    Export citation  
     
    Bookmark   3 citations  
  42.  26
    Antinomicity and the Axiom of Choice. A Chapter in Antinomic Mathematics.Florencio G. Asenjo - 1996 - Logic and Logical Philosophy 4:53-95.
    The present work is an attempt to break ground in mathematics proper, armed with the accepting view just described. Specifically, we shall examine various versions of antinomic set theory, in particular the axiom of choice, keeping the presentation as intuitive as possible, more in the manner of a nineteenth century paper than as a thoroughly formalized system. The reason for such a presentation is the conviction that at this point it should be the mathematics that eventually determines the (...)
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  43. Properties and Predicates, Objects and Names : Impredicativity and the Axiom of Choice.Stewart Shapiro - 2018 - In Ivette Fred Rivera & Jessica Leech (eds.), Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale. Oxford University Press.
     
    Export citation  
     
    Bookmark  
  44.  5
    Why the Axiom of Choice Sometimes Fails.Ivonne Victoria Pallares-Vega - 2020 - Logic Journal of the IGPL 28 (6):1207-1217.
    The early controversies surrounding the axiom of choice are well known, as are the many results that followed concerning its dependence from, and equivalence to, other mathematical propositions. This paper focuses not on the logical status of the axiom but rather on showing why it fails in certain categories.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  45.  66
    Plural Quantification and the Axiom of Choice.Stephen Pollard - 1988 - Philosophical Studies 54 (3):393 - 397.
  46.  4
    Nielsen-Schreier and the Axiom of Choice.Philipp Kleppmann - 2015 - Mathematical Logic Quarterly 61 (6):458-465.
  47.  24
    Typical Ambiguity and the Axiom of Choice.Marcel Crabbé - 1984 - Journal of Symbolic Logic 49 (4):1074-1078.
  48.  6
    Algebraic Completion Without the Axiom of Choice.Jørgen Harmse - 2022 - Mathematical Logic Quarterly 68 (4):394-397.
  49.  10
    Jourdain, Russell and the Axiom of Choice: A New Document.I. Grattan-Guinness - 2014 - Russell: The Journal of Bertrand Russell Studies 32 (1).
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  50.  45
    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.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   3 citations  
1 — 50 / 1000