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Paul Howard [31]Paul E. Howard [13]Paul Isaac Howard [1]
  1. Definitions of Compact.Paul E. Howard - 1990 - Journal of Symbolic Logic 55 (2):645-655.
  2. Definitions of Compactness and the Axiom of Choice.Omar De la Cruz, Eric Hall, Paul Howard, Jean E. Rubin & Adrienne Stanley - 2002 - 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.  26
    Products of Compact Spaces and the Axiom of Choice.O. De la Cruz, Paul Howard & E. Hall - 2002 - 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.  15
    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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  5.  16
    A Proof of a Theorem of Tennenbaum.Paul E. Howard - 1972 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 18 (7):111-112.
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  6.  4
    Compactness in Countable Tychonoff Products and Choice.Paul Howard, K. Keremedis & J. E. Rubin - 2000 - 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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  7.  9
    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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  8.  13
    Products of Compact Spaces and the Axiom of Choice II.Omar De la Cruz, Eric Hall, Paul Howard, Kyriakos Keremedis & Jean E. Rubin - 2003 - 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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  9.  14
    Russell's Alternative to the Axiom of Choice.Norbert Brunner & Paul Howard - 1992 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1):529-534.
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  10.  5
    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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  11.  14
    Non-Constructive Properties of the Real Numbers.J. E. Rubin, K. Keremedis & Paul Howard - 2001 - 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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  12.  14
    Rado's Selection Lemma Does Not Imply the Boolean Prime Ideal Theorem.Paul E. Howard - 1984 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 30 (9-11):129-132.
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  13.  14
    The Existence of Level Sets in a Free Group Implies the Axiom of Choice.Paul E. Howard - 1987 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (4):315-316.
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  14. The Fraenkel-Mostowski Method for Independence Proofs in Set Theory.J. W. Addison, Leon Henkin, Alfred Tarski & Paul E. Howard - 1975 - Journal of Symbolic Logic 40 (4):631-631.
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  15.  8
    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.
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  16.  9
    Limitations on the Fraenkel-Mostowski Method of Independence Proofs.Paul E. Howard - 1973 - 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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  17.  20
    Subgroups of a Free Group and the Axiom of Choice.Paul E. Howard - 1985 - Journal of Symbolic Logic 50 (2):458-467.
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  18.  12
    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 that the countable union (...)
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  19.  4
    No Decreasing Sequence of Cardinals.Paul Howard & Eleftherios Tachtsis - forthcoming - Archive for Mathematical Logic.
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  20.  2
    Rado's Selection Lemma Does Not Imply the Boolean Prime Ideal Theorem.Paul E. Howard - 1984 - Mathematical Logic Quarterly 30 (9‐11):129-132.
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  21.  3
    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.  9
    Versions of Normality and Some Weak Forms of the Axiom of Choice.Paul Howard, Kyriakos Keremedis, Herman Rubin & Jean E. Rubin - 1998 - 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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  23.  6
    Disjoint Unions of Topological Spaces and Choice.Paul Howard, Kyriakos Keremedis, Herman Rubin & Jean E. Rubin - 1998 - 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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  24.  22
    Independence Results for Class Forms of the Axiom of Choice.Paul E. Howard, Arthur L. Rubin & Jean E. Rubin - 1978 - 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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  25.  9
    Russell's Alternative to the Axiom of Choice.Norbert Brunner & Paul Howard - 1992 - 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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  26.  8
    Divisibility of Dedekind Finite Sets.David Blair, Andreas Blass & Paul Howard - 2005 - Journal of Mathematical Logic 5 (01):49-85.
  27.  7
    Von Rimscha's Transitivity Conditions.Paul Howard, J. E. Rubin & A. Stanley - 2000 - 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.  6
    The Boolean Prime Ideal Theorem Plus Countable Choice Do Not Imply Dependent Choice.Paul Howard & Jean E. Rubin - 1996 - 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.  19
    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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  30.  6
    Maximal $P$-Subgroups and the Axiom of Choice.Paul E. Howard & Mary Yorke - 1987 - Notre Dame Journal of Formal Logic 28 (2):276-283.
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  31.  4
    Paracompactness of Metric Spaces and the Axiom of Multiple Choice.Paul Howard, K. Keremedis & J. E. Rubin - 2000 - 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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  32.  10
    The Strength of the $\Delta$-System Lemma.Paul Howard & Jeffrey Solski - 1992 - Notre Dame Journal of Formal Logic 34 (1):100-106.
  33.  2
    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.
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  34.  2
    On Vector Spaces Over Specific Fields Without Choice.Paul Howard & Eleftherios Tachtsis - 2013 - Mathematical Logic Quarterly 59 (3):128-146.
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  35.  5
    Well Ordered Subsets of Linearly Ordered Sets.Hartmut Höft & Paul Howard - 1994 - 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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  36.  2
    The Existence of Level Sets in a Free Group Implies the Axiom of Choice.Paul E. Howard - 1987 - Mathematical Logic Quarterly 33 (4):315-316.
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  37.  1
    If Vector Spaces Are Projective Modules Then Multiple Choice Holds.Paul Howard - 2005 - 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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  38.  1
    Variations of Rado's Lemma.Paul Howard - 1993 - 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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  39.  1
    A Graph Theoretic Equivalent to the Axiom of Choice.Hartmut Höft & Paul Howard - 1973 - Mathematical Logic Quarterly 19 (11‐12):191-191.
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  40.  1
    A Proof of a Theorem of Tennenbaum.Paul E. Howard - 1972 - Mathematical Logic Quarterly 18 (7):111-112.
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  41.  1
    The Finiteness of Compact Boolean Algebras.Paul Howard - 2011 - 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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