22 found
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  1.  4
    On Ramsey’s Theorem and the Existence of Infinite Chains or Infinite Anti-Chains in Infinite Posets.Eleftherios Tachtsis - 2016 - Journal of Symbolic Logic 81 (1):384-394.
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  2.  10
    On the Minimal Cover Property and Certain Notions of Finite.Eleftherios Tachtsis - 2018 - Archive for Mathematical Logic 57 (5-6):665-686.
    In set theory without the axiom of choice, we investigate the deductive strength of the principle “every topological space with the minimal cover property is compact”, and its relationship with certain notions of finite as well as with properties of linearly ordered sets and partially ordered sets.
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  3.  5
    The Existence of Free Ultrafilters on Ω Does Not Imply the Extension of Filters on Ω to Ultrafilters.Eric J. Hall, Kyriakos Keremedis & Eleftherios Tachtsis - 2013 - Mathematical Logic Quarterly 59 (4-5):258-267.
  4.  3
    On Ramsey Choice and Partial Choice for Infinite Families of N -Element Sets.Lorenz Halbeisen & Eleftherios Tachtsis - forthcoming - Archive for Mathematical Logic:1-24.
    For an integer \, Ramsey Choice\ is the weak choice principle “every infinite setxhas an infinite subset y such that\ has a choice function”, and \ is the weak choice principle “every infinite family of n-element sets has an infinite subfamily with a choice function”. In 1995, Montenegro showed that for \, \. However, the question of whether or not \ for \ is still open. In general, for distinct \, not even the status of “\” or “\” is known. (...)
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  5.  19
    Products of Some Special Compact Spaces and Restricted Forms of AC.Kyriakos Keremedis & Eleftherios Tachtsis - 2010 - Journal of Symbolic Logic 75 (3):996-1006.
    We establish the following results: 1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent: (a) The Tychonoff product of| α| many non-empty finite discrete subsets of I is compact. (b) The union of| α| many non-empty finite subsets of I is well orderable. 2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1] (...)
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  6.  10
    On Vector Spaces Over Specific Fields Without Choice.Paul Howard & Eleftherios Tachtsis - 2013 - Mathematical Logic Quarterly 59 (3):128-146.
  7.  3
    Łoś's Theorem and the Axiom of Choice.Eleftherios Tachtsis - 2019 - Mathematical Logic Quarterly 65 (3):280-292.
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  8.  40
    Properties of the Real Line and Weak Forms of the Axiom of Choice.Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis & Eleftherios Tachtsis - 2005 - Mathematical Logic Quarterly 51 (6):598-609.
    We investigate, within the framework of Zermelo-Fraenkel set theory ZF, the interrelations between weak forms of the Axiom of Choice AC restricted to sets of reals.
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  9.  18
    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 union has (...)
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  10.  1
    On Ramsey Choice and Partial Choice for Infinite Families of N -Element Sets.Lorenz Halbeisen & Eleftherios Tachtsis - forthcoming - Archive for Mathematical Logic:1-24.
    For an integer \, Ramsey Choice\ is the weak choice principle “every infinite setxhas an infinite subset y such that\ has a choice function”, and \ is the weak choice principle “every infinite family of n-element sets has an infinite subfamily with a choice function”. In 1995, Montenegro showed that for \, \. However, the question of whether or not \ for \ is still open. In general, for distinct \, not even the status of “\” or “\” is known. (...)
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  11.  11
    A Note on the Deductive Strength of the Nielsen-Schreier Theorem.Eleftherios Tachtsis - 2018 - Mathematical Logic Quarterly 64 (3):173-177.
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  12.  9
    On Sequentially Compact Subspaces Of.Kyriakos Keremedis & Eleftherios Tachtsis - 2003 - Notre Dame Journal of Formal Logic 44 (3):175-184.
    We show that the property of sequential compactness for subspaces of.
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  13.  16
    On Infinite-Dimensional Banach Spaces and Weak Forms of the Axiom of Choice.Paul Howard & Eleftherios Tachtsis - 2017 - Mathematical Logic Quarterly 63 (6):509-535.
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  14.  7
    On Sequentially Compact Subspaces of Without the Axiom of Choice.Kyriakos Keremedis & Eleftherios Tachtsis - 2003 - Notre Dame Journal of Formal Logic 44 (3):175-184.
    We show that the property of sequential compactness for subspaces of.
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  15.  6
    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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  16.  6
    On the Set-Theoretic Strength of Ellis’ Theorem and the Existence of Free Idempotent Ultrafilters on Ω.Eleftherios Tachtsis - 2018 - Journal of Symbolic Logic 83 (2):551-571.
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  17.  15
    No Decreasing Sequence of Cardinals.Paul Howard & Eleftherios Tachtsis - 2016 - Archive for Mathematical Logic 55 (3-4):415-429.
    In set theory without the Axiom of Choice, we investigate the set-theoretic strength of the principle NDS which states that there is no function f on the set ω of natural numbers such that for everyn ∈ ω, f ≺ f, where for sets x and y, x ≺ y means that there is a one-to-one map g : x → y, but no one-to-one map h : y → x. It is a long standing open problem whether NDS implies (...)
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  18.  19
    Nonconstructive Properties of Well-Ordered T 2 Topological Spaces.Kyriakos Keremedis & Eleftherios Tachtsis - 1999 - Notre Dame Journal of Formal Logic 40 (4):548-553.
    We show that none of the following statements is provable in Zermelo-Fraenkel set theory (ZF) answering the corresponding open questions from Brunner in ``The axiom of choice in topology'':(i) For every T2 topological space (X, T) if X is well-ordered, then X has a well-ordered base,(ii) For every T2 topological space (X, T), if X is well-ordered, then there exists a function f : X × W T such that W is a well-ordered set and f ({x} × W) is (...)
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  19.  15
    Countable Sums and Products of Metrizable Spaces in ZF.Kyriakos Keremedis & Eleftherios Tachtsis - 2005 - Mathematical Logic Quarterly 51 (1):95-103.
    We study the role that the axiom of choice plays in Tychonoff's product theorem restricted to countable families of compact, as well as, Lindelöf metric spaces, and in disjoint topological unions of countably many such spaces.
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  20.  7
    On a Variant of Rado’s Selection Lemma and its Equivalence with the Boolean Prime Ideal Theorem.Paul Howard & Eleftherios Tachtsis - 2014 - Archive for Mathematical Logic 53 (7-8):825-833.
    We establish that, in ZF, the statementRLT: Given a setIand a non-empty setF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{F}}$$\end{document}of non-empty elementary closed subsets of 2Isatisfying the fip, ifF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{F}}$$\end{document}has a choice function, then⋂F≠∅\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bigcap\mathcal{F} \ne \emptyset}$$\end{document},which was introduced in Morillon :739–749, 2012), is equivalent to the Boolean Prime Ideal Theorem. The result provides, on one hand, an affirmative answer to Morillon’s corresponding (...)
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  21.  6
    Finiteness Classes and Small Violations of Choice.Horst Herrlich, Paul Howard & Eleftherios Tachtsis - 2016 - Notre Dame Journal of Formal Logic 57 (3):375-388.
    We study properties of certain subclasses of the Dedekind finite sets in set theory without the axiom of choice with respect to the comparability of their elements and to the boundedness of such classes, and we answer related open problems from Herrlich’s “The Finite and the Infinite.” The main results are as follows: 1. It is relatively consistent with ZF that the class of all finite sets is not the only finiteness class such that any two of its elements are (...)
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  22.  3
    Odd-Sized Partitions of Russell-Sets.Horst Herrlich & Eleftherios Tachtsis - 2010 - Mathematical Logic Quarterly 56 (2):185-190.
    In the setting of ZF, i.e., Zermelo–Fraenkel set theory without the Axiom of Choice , we study partitions of Russell-sets into sets each with exactly n elements , for some integer n. We show that if n is odd, then a Russell-set X has an n -ary partition if and only if |X | is divisible by n. Furthermore, we establish that it is relative consistent with ZF that there exists a Russell-set X such that |X | is not divisible (...)
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