26 found
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  1.  10
    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.  16
    On Ramsey choice and partial choice for infinite families of n -element sets.Lorenz Halbeisen & Eleftherios Tachtsis - 2020 - Archive for Mathematical Logic 59 (5-6):583-606.
    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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  3.  5
    MA(ℵ0) restricted to complete Boolean algebras and choice.Eleftherios Tachtsis - 2021 - Mathematical Logic Quarterly 67 (4):420-431.
    It is a long standing open problem whether or not the Axiom of Countable Choice implies the fragment of Martin's Axiom either in or in. In this direction, we provide a partial answer by establishing that the Boolean Prime Ideal Theorem in conjunction with the Countable Union Theorem does not imply restricted to complete Boolean algebras in. Furthermore, we prove that the latter (formally) weaker form of and the Δ‐system Lemma are independent of each other in.We also answer open questions (...)
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  4.  18
    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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  5.  18
    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.
    Let X be an infinite set and let and denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by the Stone space of the Boolean algebra of all subsets of X. We show: For every well‐ordered cardinal number ℵ, (ℵ) iff (2ℵ). iff “ is a continuous image of ” iff “ has a free open ultrafilter ” iff “every countably infinite subset of has a limit point”. (...)
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  6.  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 union has (...)
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  7.  25
    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.
    We study theorems from Functional Analysis with regard to their relationship with various weak choice principles and prove several results about them: “Every infinite‐dimensional Banach space has a well‐orderable Hamel basis” is equivalent to ; “ can be well‐ordered” implies “no infinite‐dimensional Banach space has a Hamel basis of cardinality ”, thus the latter statement is true in every Fraenkel‐Mostowski model of ; “No infinite‐dimensional Banach space has a Hamel basis of cardinality ” is not provable in ; “No infinite‐dimensional (...)
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  8.  37
    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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  9.  39
    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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  10.  16
    Countable products and countable direct sums of compact metrizable spaces in the absence of the Axiom of Choice.Kyriakos Keremedis, Eleftherios Tachtsis & Eliza Wajch - 2023 - Annals of Pure and Applied Logic 174 (7):103283.
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  11.  13
    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 these (...)
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  12.  11
    On Hausdorff operators in ZF$\mathsf {ZF}$.Kyriakos Keremedis & Eleftherios Tachtsis - 2023 - Mathematical Logic Quarterly 69 (3):347-369.
    A Hausdorff space is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every with,, where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in, i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice (): is equivalent to “For every set X, the Cantor cube is effectively Hausdorff”. This enhances the result of Howard, Keremedis, Rubin and Rubin [13] that is equivalent to “Hausdorff (...)
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  13.  21
    On vector spaces over specific fields without choice.Paul Howard & Eleftherios Tachtsis - 2013 - Mathematical Logic Quarterly 59 (3):128-146.
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  14.  55
    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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  15.  16
    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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  16.  12
    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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  17.  16
    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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  18.  25
    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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  19.  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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  20.  15
    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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  21.  14
    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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  22.  33
    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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  23.  15
    Almost Disjoint and Mad Families in Vector Spaces and Choice Principles.Eleftherios Tachtsis - 2022 - Journal of Symbolic Logic 87 (3):1093-1110.
    In set theory without the Axiom of Choice ( $\mathsf {AC}$ ), we investigate the open problem of the deductive strength of statements which concern the existence of almost disjoint and maximal almost disjoint (MAD) families of infinite-dimensional subspaces of a given infinite-dimensional vector space, as well as the extension of almost disjoint families in infinite-dimensional vector spaces to MAD families.
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  24.  19
    A note on the deductive strength of the Nielsen‐Schreier theorem.Eleftherios Tachtsis - 2018 - Mathematical Logic Quarterly 64 (3):173-177.
    We show that the Boolean Prime Ideal Theorem () does not imply the Nielsen‐Schreier Theorem () in, thus strengthening the result of Kleppmann from “Nielsen‐Schreier and the Axiom of Choice” that the (strictly weaker than ) Ordering Principle () does not imply in. We also show that is false in Mostowski's Linearly Ordered Model of. The above two results also settle the corresponding open problems from Howard and Rubin's “Consequences of the Axiom of Choice”.
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  25.  18
    Łoś's theorem and the axiom of choice.Eleftherios Tachtsis - 2019 - Mathematical Logic Quarterly 65 (3):280-292.
    In set theory without the Axiom of Choice (), we investigate the problem of the placement of Łoś's Theorem () in the hierarchy of weak choice principles, and answer several open questions from the book Consequences of the Axiom of Choice by Howard and Rubin, as well as an open question by Brunner. We prove a number of results summarised in § 3.
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  26.  13
    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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