41 found
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Paul Howard [30]Paul E. Howard [13]Paul Isaac Howard [1]
  1. Paul E. Howard (1990). Definitions of Compact. Journal of Symbolic Logic 55 (2):645-655.
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  2. Omar De la Cruz, Eric Hall, Paul Howard, Jean E. Rubin & Adrienne Stanley (2002). Definitions of Compactness and the Axiom of Choice. Journal of Symbolic Logic 67 (1):143-161.
    We study the relationships between definitions of compactness in topological spaces and the roll the axiom of choice plays in these relationships.
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  3.  23
    O. De la Cruz, Paul Howard & E. Hall (2002). Products of Compact Spaces and the Axiom of Choice. Mathematical Logic Quarterly 48 (4):508-516.
    We study the Tychonoff Compactness Theorem for several different definitions of a compact space.
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  4.  3
    Paul Howard & Eleftherios Tachtsis (forthcoming). No Decreasing Sequence of Cardinals. Archive for Mathematical Logic.
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  5.  14
    Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis & Eleftherios Tachtsis (2005). Properties of the Real Line and Weak Forms of the Axiom of Choice. 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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  6.  5
    Norbert Brunner & Paul Howard (1992). Russell's Alternative to the Axiom of Choice. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1):529-534.
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  7.  7
    Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis & Jean E. Rubin (2003). Metric Spaces and the Axiom of Choice. 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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  8.  13
    J. E. Rubin, K. Keremedis & Paul Howard (2001). Non-Constructive Properties of the Real Numbers. Mathematical Logic Quarterly 47 (3):423-431.
    We study the relationship between various properties of the real numbers and weak choice principles.
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  9.  2
    Horst Herrlich, Paul Howard & Eleftherios Tachtsis (forthcoming). Finiteness Classes and Small Violations of Choice. Notre Dame Journal of Formal Logic.
    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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  10.  4
    Paul Howard, K. Keremedis & J. E. Rubin (2000). Compactness in Countable Tychonoff Products and Choice. Mathematical Logic Quarterly 46 (1):3-16.
    We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.
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  11.  4
    Paul E. Howard (1972). A Proof of a Theorem of Tennenbaum. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 18 (7):111-112.
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  12.  4
    Paul Howard (2007). Bases, Spanning Sets, and the Axiom of Choice. 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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  13.  4
    Paul E. Howard (1984). Rado's Selection Lemma Does Not Imply the Boolean Prime Ideal Theorem. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 30 (9-11):129-132.
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  14. J. W. Addison, Leon Henkin, Alfred Tarski & Paul E. Howard (1975). The Fraenkel-Mostowski Method for Independence Proofs in Set Theory. Journal of Symbolic Logic 40 (4):631-631.
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  15.  11
    Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis & Jean E. Rubin (2003). Products of Compact Spaces and the Axiom of Choice II. 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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  16.  9
    Omar De la Cruz, Eric J. Hall, Paul Howard, Kyriakos Keremedis & Jean E. Rubin (2008). Unions and the Axiom of Choice. 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 that the countable union (...)
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  17.  4
    Paul E. Howard (1992). The Axiom of Choice for Countable Collections of Countable Sets Does Not Imply the Countable Union Theorem. Notre Dame Journal of Formal Logic 33 (2):236-243.
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  18.  5
    Paul Howard, Kyriakos Keremedis, Herman Rubin & Jean E. Rubin (1998). Disjoint Unions of Topological Spaces and Choice. Mathematical Logic Quarterly 44 (4):493-508.
    We find properties of topological spaces which are not shared by disjoint unions in the absence of some form of the Axiom of Choice.
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  19.  9
    Paul Howard, Kyriakos Keremedis, Herman Rubin & Jean E. Rubin (1998). Versions of Normality and Some Weak Forms of the Axiom of Choice. Mathematical Logic Quarterly 44 (3):367-382.
    We investigate the set theoretical strength of some properties of normality, including Urysohn's Lemma, Tietze-Urysohn Extension Theorem, normality of disjoint unions of normal spaces, and normality of Fσ subsets of normal spaces.
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  20.  9
    Paul E. Howard (1973). Limitations on the Fraenkel-Mostowski Method of Independence Proofs. Journal of Symbolic Logic 38 (3):416-422.
    The Fraenkel-Mostowski method has been widely used to prove independence results among weak versions of the axiom of choice. In this paper it is shown that certain statements cannot be proved by this method. More specifically it is shown that in all Fraenkel-Mostowski models the following hold: 1. The axiom of choice for sets of finite sets implies the axiom of choice for sets of well-orderable sets. 2. The Boolean prime ideal theorem implies a weakened form of Sikorski's theorem.
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  21.  8
    David Blair, Andreas Blass & Paul Howard (2005). Divisibility of Dedekind Finite Sets. Journal of Mathematical Logic 5 (01):49-85.
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  22.  13
    Paul E. Howard (1985). Subgroups of a Free Group and the Axiom of Choice. Journal of Symbolic Logic 50 (2):458-467.
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  23.  2
    Paul Howard & Eleftherios Tachtsis (2014). On a Variant of Rado’s Selection Lemma and its Equivalence with the Boolean Prime Ideal Theorem. Archive for Mathematical Logic 53 (7-8):825-833.
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  24.  8
    Norbert Brunner & Paul Howard (1992). Russell's Alternative to the Axiom of Choice. Mathematical Logic Quarterly 38 (1):529-534.
    We prove the independence of some weakenings of the axiom of choice related to the question if the unions of wellorderable families of wellordered sets are wellorderable.
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  25.  2
    Paul E. Howard (1987). The Existence of Level Sets in a Free Group Implies the Axiom of Choice. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (4):315-316.
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  26.  1
    Paul E. Howard (1984). Rado's Selection Lemma Does Not Imply the Boolean Prime Ideal Theorem. Mathematical Logic Quarterly 30 (9‐11):129-132.
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  27.  6
    Paul Howard, J. E. Rubin & A. Stanley (2000). Von Rimscha's Transitivity Conditions. Mathematical Logic Quarterly 46 (4):549-554.
    In Zermelo-Fraenkel set theory with the axiom of choice every set has the same cardinal number as some ordinal. Von Rimscha has weakened this condition to “Every set has the same cardinal number as some transitive set”. In set theory without the axiom of choice, we study the deductive strength of this and similar statements introduced by von Rimscha.
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  28.  5
    Paul Howard & Jean E. Rubin (1996). The Boolean Prime Ideal Theorem Plus Countable Choice Do Not Imply Dependent Choice. Mathematical Logic Quarterly 42 (1):410-420.
    Two Fraenkel-Mostowski models are constructed in which the Boolean Prime Ideal Theorem is true. In both models, AC for countable sets is true, but AC for sets of cardinality 2math image and the 2m = m principle are both false. The Principle of Dependent Choices is true in the first model, but false in the second.
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  29.  3
    Paul Howard, K. Keremedis & J. E. Rubin (2000). Paracompactness of Metric Spaces and the Axiom of Multiple Choice. 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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  30.  18
    Paul E. Howard, Arthur L. Rubin & Jean E. Rubin (1978). Independence Results for Class Forms of the Axiom of Choice. Journal of Symbolic Logic 43 (4):673-684.
    Let NBG be von Neumann-Bernays-Gödel set theory without the axiom of choice and let NBGA be the modification which allows atoms. In this paper we consider some of the well-known class or global forms of the wellordering theorem, the axiom of choice, and maximal principles which are known to be equivalent in NBG and show they are not equivalent in NBGA.
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  31.  16
    Paul Howard & Jean E. Rubin (1995). The Axiom of Choice for Well-Ordered Families and for Families of Well- Orderable Sets. 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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  32.  4
    Paul E. Howard & Mary Yorke (1987). Maximal $P$-Subgroups and the Axiom of Choice. Notre Dame Journal of Formal Logic 28 (2):276-283.
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  33.  10
    Paul Howard & Jeffrey Solski (1992). The Strength of the $\Delta$-System Lemma. Notre Dame Journal of Formal Logic 34 (1):100-106.
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  34.  4
    Hartmut Höft & Paul Howard (1994). Well Ordered Subsets of Linearly Ordered Sets. Notre Dame Journal of Formal Logic 35 (3):413-425.
    The deductive relationships between six statements are examined in set theory without the axiom of choice. Each of these statements follows from the axiom of choice and involves linear orderings in some way.
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  35.  1
    Paul Howard (1993). Variations of Rado's Lemma. Mathematical Logic Quarterly 39 (1):353-356.
    The deductive strengths of three variations of Rado's selection lemma are studied in set theory without the axiom of choice. Two are shown to be equivalent to Rado's lemma and the third to the Boolean prime ideal theorem. MSC: 03E25, 04A25, 06E05.
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  36.  1
    Paul Howard & Eleftherios Tachtsis (2013). On Vector Spaces Over Specific Fields Without Choice. Mathematical Logic Quarterly 59 (3):128-146.
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  37.  1
    Hartmut Höft & Paul Howard (1973). A Graph Theoretic Equivalent to the Axiom of Choice. Mathematical Logic Quarterly 19 (11‐12):191-191.
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  38.  1
    Paul E. Howard (1972). A Proof of a Theorem of Tennenbaum. Mathematical Logic Quarterly 18 (7):111-112.
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  39.  1
    Paul E. Howard (1987). The Existence of Level Sets in a Free Group Implies the Axiom of Choice. Mathematical Logic Quarterly 33 (4):315-316.
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  40.  1
    Paul Howard (2011). The Finiteness of Compact Boolean Algebras. Mathematical Logic Quarterly 57 (1):14-18.
    We show that it consistent with Zermelo-Fraenkel set theory that there is an infinite, compact Boolean algebra.
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  41. Paul Howard (2005). If Vector Spaces Are Projective Modules Then Multiple Choice Holds. 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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