Results for 'Axiom of choice'

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  1.  6
    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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  2. 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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  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.
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  4.  98
    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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  5.  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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  6.  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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  7.  48
    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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  8. 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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  9.  63
    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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  10.  43
    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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  11.  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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  12.  22
    Axiom of Choice for Finite Sets.Andrzej Mostowski - 1948 - Journal of Symbolic Logic 13 (1):45-46.
  13.  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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  14. The Axiom of choice in Quine's New Foundations for Mathematical Logic.Ernst P. Specker - 1954 - Journal of Symbolic Logic 19 (2):127-128.
  15.  46
    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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  16.  80
    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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  17. 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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  18.  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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  19.  21
    Axiom of choice and excluded middle in categorical logic.Steven Awodey - 1995 - Bulletin of Symbolic Logic 1:344.
  20.  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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  21.  10
    The Axiom of Choice and the Class of Hyperarithmetic Functions.G. Kreisel - 1970 - Journal of Symbolic Logic 35 (2):333-334.
  22.  16
    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.  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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  24. The Axiom of Choice Vol. 22.John L. Bell - 2009 - College Publications.
     
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  25.  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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  26.  23
    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.
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  27.  36
    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.
  28.  14
    Nielsen‐Schreier and the Axiom of Choice.Philipp Kleppmann - 2015 - Mathematical Logic Quarterly 61 (6):458-465.
    The Nielsen‐Schreier theorem asserts that subgroups of free groups are free. In the first section we show that this theorem does not follow from the Linear Ordering Principle, thus strengthening the fact that it implies the Axiom of Choice for families of finite sets. In the second section, we show that a stronger variant of the Nielsen‐Schreier theorem implies the Axiom of Choice.
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  29.  46
    Zermelo's Axiom of Choice. Its Origins, Development, and Influence.Gregory H. Moore - 1984 - Journal of Symbolic Logic 49 (2):659-660.
  30.  39
    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 (...)
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  31.  16
    Consequences of the failure of the axiom of choice in the theory of Lindelof metric spaces.Kyriakos Keremedis - 2004 - Mathematical Logic Quarterly 50 (2):141.
    We study within the framework of Zermelo-Fraenkel set theory ZF the role that the axiom of choice plays in the theory of Lindelöf metric spaces. We show that in ZF the weak choice principles: Every Lindelöf metric space is separable and Every Lindelöf metric space is second countable are equivalent. We also prove that a Lindelöf metric space is hereditarily separable iff it is hereditarily Lindelöf iff it hold as well the axiom of choice restricted (...)
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  32.  34
    Extending Independent Sets to Bases and the Axiom of Choice.Kyriakos Keremedis - 1998 - Mathematical Logic Quarterly 44 (1):92-98.
    We show that the both assertions “in every vector space B over a finite element field every subspace V ⊆ B has a complementary subspace S” and “for every family [MATHEMATICAL SCRIPT CAPITAL A] of disjoint odd sized sets there exists a subfamily ℱ={Fj:j ϵω} with a choice function” together imply the axiom of choice AC. We also show that AC is equivalent to the statement “in every vector space over ℚ every generating set includes a basis”.
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  33.  40
    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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  34.  29
    Some Weak Forms of the Axiom of Choice Restricted to the Real Line.Kyriakos Keremedis & Eleftherios Tachtsis - 2001 - Mathematical Logic Quarterly 47 (3):413-422.
    It is shown that AC, the axiom of choice for families of non-empty subsets of the real line ℝ, does not imply the statement PW, the powerset of ℝ can be well ordered. It is also shown that the statement “the set of all denumerable subsets of ℝ has size 2math image” is strictly weaker than AC and each of the statements “if every member of an infinite set of cardinality 2math image has power 2math image, then the (...)
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  35.  70
    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 (...)
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  36.  17
    Hintikka’s Take on the Axiom of Choice and the Constructivist Challenge.Radmila Jovanović - 2013 - Revista de Humanidades de Valparaíso 2:135-150.
    In the present paper we confront Martin- Löf’s analysis of the axiom of choice with J. Hintikka’s standing on this axiom. Hintikka claims that his game theoretical semantics for Independence Friendly Logic justifies Zermelo’s axiom of choice in a first-order way perfectly acceptable for the constructivists. In fact, Martin- Löf’s results lead to the following considerations:Hintikka preferred version of the axiom of choice is indeed acceptable for the constructivists and its meaning does not (...)
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  37.  31
    Bases, spanning sets, and the axiom of choice.Paul Howard - 2007 - Mathematical Logic Quarterly 53 (3):247-254.
    Two theorems are proved: First that the statement“there exists a field F such that for every vector space over F, every generating set contains a basis”implies the axiom of choice. This generalizes theorems of Halpern, Blass, and Keremedis. Secondly, we prove that the assertion that every vector space over ℤ2 has a basis implies that every well-ordered collection of two-element sets has a choice function.
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  38.  45
    The axiom of choice in topology.Norbert Brunner - 1983 - Notre Dame Journal of Formal Logic 24 (3):305-317.
  39. 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;: Princeton University Press;. Edited by George William Brown.
    Kurt Gödel, mathematician and logician, was one of the most influential thinkers of the twentieth century. Gödel fled Nazi Germany, fearing for his Jewish wife and fed up with Nazi interference in the affairs of the mathematics institute at the University of Göttingen. In 1933 he settled at the Institute for Advanced Study in Princeton, where he joined the group of world-famous mathematicians who made up its original faculty. His 1940 book, better known by its short title, The Consistency of (...)
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  40.  59
    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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  41.  75
    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 (...)
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  42.  29
    Two topological equivalents of the axiom of choice.Eric Schechter & E. Schechter - 1992 - Mathematical Logic Quarterly 38 (1):555-557.
    We show that the Axiom of Choice is equivalent to each of the following statements: A product of closures of subsets of topological spaces is equal to the closure of their product ; A product of complete uniform spaces is complete.
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  43.  11
    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.
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  44. 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 (...)
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  45.  89
    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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  46. Assertion Proof and the Axiom of Choice.David DeVidi - 2006 - In ¸ Itedevidikenyon2006. Springer Verlag.
     
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  47.  15
    On Lindelof Metric Spaces and Weak Forms of the Axiom of Choice.Kyriakos Keremedis & Eleftherios Tachtsis - 2000 - Mathematical Logic Quarterly 46 (1):35-44.
    We show that the countable axiom of choice CAC strictly implies the statements “Lindelöf metric spaces are second countable” “Lindelöf metric spaces are separable”. We also show that CAC is equivalent to the statement: “If is a Lindelöf topological space with respect to the base ℬ, then is Lindelöf”.
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  48.  87
    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 (...)
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  49. 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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  50.  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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