Search results for 's4 axiom' (try it on Scholar)

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  1. Susanne Bobzien (2012). If It's Clear, Then It's Clear That It's Clear, or is It? Higher-Order Vagueness and the S4 Axiom. In B. Morison K. Ierodiakonou (ed.), Episteme, etc. OUP UK.score: 186.0
    The purpose of this paper is to challenge some widespread assumptions about the role of the modal axiom 4 in a theory of vagueness. In the context of vagueness, axiom 4 usually appears as the principle ‘If it is clear (determinate, definite) that A, then it is clear (determinate, definite) that it is clear (determinate, definite) that A’, or, more formally, CA → CCA. We show how in the debate over axiom 4 two different notions of clarity (...)
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  2. Timothy Williamson (1995). Does Assertibility Satisfy the S4 Axiom? Critica 27 (81):3 - 25.score: 150.0
    N. B. Prof Williamson is now based at the Faculty of Philosophy, University of Oxford.
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  3. Leo Simons (1962). A Reduction in the Number of Independent Axiom Schemata for $S4$. Notre Dame Journal of Formal Logic 3 (4):256-258.score: 120.0
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  4. R. I. Goldblatt (1973). Concerning the Proper Axiom for $S4.04$ and Some Related Systems. Notre Dame Journal of Formal Logic 14 (3):392-396.score: 120.0
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  5. Naoto Yonemitsu (1967). Review: Leo Simons, A Reduction in the Number of Independent Axiom Schemata for S4. [REVIEW] Journal of Symbolic Logic 32 (2):245-245.score: 120.0
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  6. R. I. Goldblatt (1975). Erratum: ``Concerning the Proper Axiom for $S4.04$ and Some Related Systems''. Notre Dame Journal of Formal Logic 16 (4):608-608.score: 120.0
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  7. Thomas Mormann, McKinsey Algebras and Topological Models of S4.1.score: 102.0
    The aim of this paper is to show that every topological space gives rise to a wealth of topological models of the modal logic S4.1. The construction of these models is based on the fact that every space defines a Boolean closure algebra (to be called a McKinsey algebra) that neatly reflects the structure of the modal system S4.1. It is shown that the class of topological models based on McKinsey algebras contains a canonical model that can be used to (...)
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  8. Branden Fitelson (2002). Shortest Axiomatizations of Implicational S4 and S5. Notre Dame Journal of Formal Logic 43 (3):169-179.score: 66.0
    Shortest possible axiomatizations for the strict implicational fragments of the modal logics S4 and S5 are reported. Among these axiomatizations is included a shortest single axiom for implicational S4—which to our knowledge is the first reported single axiom for that system—and several new shortest single axioms for implicational S5. A variety of automated reasoning strategies were essential to our discoveries.
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  9. Zachary Ernst, Branden Fitelson, Kenneth Harris & Larry Wos (2002). Shortest Axiomatizations of Implicational S4 and S. Notre Dame Journal of Formal Logic 43 (3):169-179.score: 66.0
    Shortest possible axiomatizations for the implicational fragments of the modal logics S4 and S5 are reported. Among these axiomatizations is included a shortest single axiom for implicational S4—which to our knowledge is the first reported single axiom for that system—and several new shortest single axioms for implicational S5. A variety of automated reasoning strategies were essential to our discoveries.
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  10. Gemma Robles & José M. Méndez (2010). Axiomatizing S4+ and J+ Without the Suffixing, Prefixing and Self-Distribution of the Conditional Axioms. Bulletin of the Section of Logic 39 (1/2):79-91.score: 50.0
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  11. H. Montgomery & Richard Routley (1968). Non-Contingency Axioms for S4 and S5. Logique Et Analyse 11:422-424.score: 50.0
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  12. A. N. Prior (1957). Review: Arata Ishimoto, A Note on the Paper "A Set of Axioms of the Modal Propositional Calculus Equivalent to S3."; Arata Ishimoto, A Formulation of the Modal Propositional Calculus Equivalent to S4. [REVIEW] Journal of Symbolic Logic 22 (3):326-327.score: 50.0
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  13. Donald Paul Snyder (1973). Review: J. Jay Zeman, Bases for S4 and S4.2 Without Added Axioms. [REVIEW] Journal of Symbolic Logic 38 (2):328-328.score: 50.0
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  14. José M. Méndez & Gemma Robles (2007). Relevance Logics, Paradoxes of Consistency and the K Rule II. A Non-Constructive Negation. Logic and Logical Philosophy 15 (3):175-191.score: 30.0
    The logic B+ is Routley and Meyer’s basic positive logic. We define the logics BK+ and BK'+ by adding to B+ the K rule and to BK+ the characteristic S4 axiom, respectively. These logics are endowed with a relatively strong non-constructive negation. We prove that all the logics defined lack the K axiom and the standard paradoxes of consistency.
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  15. Lee Walters (2014). The Possibility of Unicorns and Modal Logic. Analytic Philosophy 55 (2):295-305.score: 24.0
    Michael Dummett argues, against Saul Kripke, that there could have been unicorns. He then claims that this possibility shows that the logic of metaphysical modality is not S5, and, in particular, that the B axiom is false. Dummett’s argument against B, however, is invalid. I show that although there are number of ways to repair Dummett’s argument against B, each requires a controversial metaphysical or semantic commitment, and that, regardless of this, the case against B is undermotivated. Dummett’s case (...)
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  16. Olivier Esser (2000). Inconsistency of the Axiom of Choice with the Positive Theory GPK+ ∞. Journal of Symbolic Logic 65 (4):1911 - 1916.score: 24.0
    The idea of the positive theory is to avoid the Russell's paradox by postulating an axiom scheme of comprehension for formulas without "too much" negations. In this paper, we show that the axiom of choice is inconsistent with the positive theory GPK + ∞.
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  17. Paul Corazza (2010). The Axiom of Infinity and Transformations J: V→V. Bulletin of Symbolic Logic 16 (1):37-84.score: 24.0
    We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought (...)
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  18. David W. Miller (2007). Some Restricted Lindenbaum Theorems Equivalent to the Axiom of Choice. Logica Universalis 1 (1):183-199.score: 24.0
    . 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 . A (...)
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  19. Gemma Robles & José M. Méndez (2014). Curry's Paradox, Generalized Modus Ponens Axiom and Depth Relevance. Studia Logica 102 (1):185-217.score: 24.0
    “Weak relevant model structures” (wr-ms) are defined on “weak relevant matrices” by generalizing Brady’s model structure ${\mathcal{M}_{\rm CL}}$ built upon Meyer’s Crystal matrix CL. It is shown how to falsify in any wr-ms the Generalized Modus Ponens axiom and similar schemes used to derive Curry’s Paradox. In the last section of the paper we discuss how to extend this method of falsification to more general schemes that could also be used in deriving Curry’s Paradox.
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  20. Peter Nyikos & Leszek Piątkiewicz (1995). On the Equivalence of Certain Consequences of the Proper Forcing Axiom. Journal of Symbolic Logic 60 (2):431-443.score: 24.0
    We prove that a number of axioms, each a consequence of PFA (the Proper Forcing Axiom) are equivalent. In particular we show that TOP (the Thinning-out Principle as introduced by Baumgartner in the Handbook of set-theoretic topology), is equivalent to the following statement: If I is an ideal on ω 1 with ω 1 generators, then there exists an uncountable $X \subseteq \omega_1$ , such that either [ X] ω ∩ I = ⊘ or $\lbrack X\rbrack^\omega \subseteq I$.
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  21. Larisa Maksimova (2002). Complexity of Interpolation and Related Problems in Positive Calculi. Journal of Symbolic Logic 67 (1):397-408.score: 24.0
    We consider the problem of recognizing important properties of logical calculi and find complexity bounds for some decidable properties. For a given logical system L, a property P of logical calculi is called decidable over L if there is an algorithm which for any finite set Ax of new axiom schemes decides whether the calculus L + Ax has the property P or not. In [11] the complexity of tabularity, pre-tabularity, and interpolation problems over the intuitionistic logic Int and (...)
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  22. W. J. Blok (1980). The Lattice of Modal Logics: An Algebraic Investigation. Journal of Symbolic Logic 45 (2):221-236.score: 24.0
    Modal logics are studied in their algebraic disguise of varieties of so-called modal algebras. This enables us to apply strong results of a universal algebraic nature, notably those obtained by B. Jonsson. It is shown that the degree of incompleteness with respect to Kripke semantics of any modal logic containing the axiom □ p → p or containing an axiom of the form $\square^mp \leftrightarrow\square^{m + 1}p$ for some natural number m is 2 ℵ 0 . Furthermore, we (...)
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  23. Samir Chopra, Aditya Ghose, Thomas Meyer & Ka-Shu Wong (2008). Iterated Belief Change and the Recovery Axiom. Journal of Philosophical Logic 37 (5):501 - 520.score: 24.0
    The axiom of recovery, while capturing a central intuition regarding belief change, has been the source of much controversy. We argue briefly against putative counterexamples to the axiom—while agreeing that some of their insight deserves to be preserved—and present additional recovery-like axioms in a framework that uses epistemic states, which encode preferences, as the object of revisions. This makes iterated revision possible and renders explicit the connection between iterated belief change and the axiom of recovery. We provide (...)
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  24. David Bennett (2000). A Single Axiom for Set Theory. Notre Dame Journal of Formal Logic 41 (2):152-170.score: 24.0
    Axioms in set theory typically have the form , where is a relation which links with in some way. In this paper we introduce a particular linkage relation and a single axiom based on from which all the axioms of (Zermelo set theory) can be derived as theorems. The single axiom is presented both in informal and formal versions. This calls for some discussion of pertinent features of formal and informal axiomatic method and some discussion of pertinent features (...)
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  25. Teruyuki Yorioka (2008). Some Weak Fragments of Martin's Axiom Related to the Rectangle Refining Property. Archive for Mathematical Logic 47 (1):79-90.score: 24.0
    We introduce the anti-rectangle refining property for forcing notions and investigate fragments of Martin’s axiom for ℵ1 dense sets related to the anti-rectangle refining property, which is close to some fragment of Martin’s axiom for ℵ1 dense sets related to the rectangle refining property, and prove that they are really weaker fragments.
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  26. Jesse Alama (2014). The Simplest Axiom System for Hyperbolic Geometry Revisited, Again. Studia Logica 102 (3):609-615.score: 24.0
    Dependencies are identified in two recently proposed first-order axiom systems for plane hyperbolic geometry. Since the dependencies do not specifically concern hyperbolic geometry, our results yield two simpler axiom systems for absolute geometry.
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  27. Paul Corazza (2000). Consistency of V = HOD with the Wholeness Axiom. Archive for Mathematical Logic 39 (3):219-226.score: 24.0
    The Wholeness Axiom (WA) is an axiom schema that can be added to the axioms of ZFC in an extended language $\{\in,j\}$ , and that asserts the existence of a nontrivial elementary embedding $j:V\to V$ . The well-known inconsistency proofs are avoided by omitting from the schema all instances of Replacement for j-formulas. We show that the theory ZFC + V = HOD + WA is consistent relative to the existence of an $I_1$ embedding. This answers a question (...)
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  28. Edwin D. Mares & Robert K. Meyer (1993). The Semantics Ofr. Journal of Philosophical Logic 22 (1):95 - 110.score: 24.0
    The Logic R4 is obtained by adding the axiom □(A v B) → (◇A v □B) to the modal relevant logic NR. We produce a model theory for this logic and show completeness. We also show that there is a natural embedding of a Kripke model for S4 in each R4 model structure.
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  29. Rajeev Gore, Advances in Modal Logic, Volume.score: 24.0
    We study a propositional bimodal logic consisting of two S4 modalities and [a], together with the interaction axiom scheme a ϕ → a ϕ. In the intended semantics, the plain..
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  30. Kyriakos Keremedis (2001). Disasters in Topology Without the Axiom of Choice. Archive for Mathematical Logic 40 (8):569-580.score: 24.0
    We show that some well known theorems in topology may not be true without the axiom of choice.
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  31. Marianne Morillon (2010). Notions of Compactness for Special Subsets of ℝ I and Some Weak Forms of the Axiom of Choice. Journal of Symbolic Logic 75 (1):255-268.score: 24.0
    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 (resp. the (...) of Dependent Choices) was required. If I is linearly orderable (for example $I={\Bbb R}$ ), then, in ZF, the closed unit ball of the Hilbert space $\ell ^{2}(I)$ is (Loeb-)compact in the weak topology. However, the weak compactness of the closed unit ball of $\ell ^{2}(\scr{P}({\Bbb R}))$ is not provable in ZF. (shrink)
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  32. Huiling Zhu (2013). Distributive Proper Forcing Axiom and Cardinal Invariants. Archive for Mathematical Logic 52 (5-6):497-506.score: 24.0
    In this paper, we study the forcing axiom for the class of proper forcing notions which do not add ω sequence of ordinals. We study the relationship between this forcing axiom and many cardinal invariants. We use typical iterated forcing with large cardinals and analyse certain property being preserved in this process. Lastly, we apply the results to distinguish several forcing axioms.
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  33. Greg Oman (2010). On the Axiom of Union. Archive for Mathematical Logic 49 (3):283-289.score: 24.0
    In this paper, we study the union axiom of ZFC. After a brief introduction, we sketch a proof of the folklore result that union is independent of the other axioms of ZFC. In the third section, we prove some results in the theory T:= ZFC minus union. Finally, we show that the consistency of T plus the existence of an inaccessible cardinal proves the consistency of ZFC.
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  34. Victor Pambuccian (2011). The Simplest Axiom System for Plane Hyperbolic Geometry Revisited. Studia Logica 97 (3):347 - 349.score: 24.0
    Using the axiom system provided by Carsten Augat in [1], it is shown that the only 6-variable statement among the axioms of the axiom system for plane hyperbolic geometry (in Tarski's language L B =), we had provided in [3], is superfluous. The resulting axiom system is the simplest possible one, in the sense that each axiom is a statement in prenex form about at most 5 points, and there is no axiom system consisting entirely (...)
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  35. F. Wolter, H. Wansing, M. de Rijke & M. Zakharyaschev, Advances in Modal Logic, Volume.score: 24.0
    We study a propositional bimodal logic consisting of two S4 modalities £ and [a], together with the interaction axiom scheme a £ϕ → £ aϕ. In the intended semantics, the plain £ is given the McKinsey-Tarski interpretation as the interior operator of a topology, while the labelled [a] is given the standard Kripke semantics using a reflexive and transitive binary relation a. The interaction axiom expresses the property that the Ra relation is lower semi-continuous with respect to the (...)
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  36. John Bell (2008). The Axiom of Choice and the Law of Excluded Middle in Weak Set Theories. Mathematical Logic Quarterly 54 (2):194-201.score: 21.0
    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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  37. M. Rathjen (2001). Kripke-Platek Set Theory and the Anti-Foundation Axiom. Mathematical Logic Quarterly 47 (4):435-440.score: 21.0
    The paper investigates the strength of the Anti-Foundation Axiom, AFA, on the basis of Kripke-Platek set theory without Foundation. It is shown that the addition of AFA considerably increases the proof theoretic strength.
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  38. 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.score: 21.0
    We study the Tychonoff Compactness Theorem for several different definitions of a compact space.
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  39. Stefano Baratella & Ruggero Ferro (1993). A Theory of Sets with the Negation of the Axiom of Infinity. Mathematical Logic Quarterly 39 (1):338-352.score: 21.0
    In this paper we introduce a theory of finite sets FST with a strong negation of the axiom of infinity asserting that every set is provably bijective with a natural number. We study in detail the role of the axioms of Power Set, Choice, Regularity in FST, pointing out the relative dependences or independences among them. FST is shown to be provably equivalent to a fragment of Alternative Set Theory. Furthermore, the introduction of FST is motivated in view of (...)
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  40. 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.score: 21.0
    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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  41. Christine Gaßner (1994). The Axiom of Choice in Second‐Order Predicate Logic. Mathematical Logic Quarterly 40 (4):533-546.score: 21.0
    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 for (...)
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  42. Wlodek Rabinowicz (1997). On Seidenfeld‘s Criticism of Sophisticated Violations of the Independence Axiom. Theory and Decision 43 (3):279-292.score: 21.0
    An agent who violates independence can avoid dynamic inconsistency in sequential choice if he is sophisticated enough to make use of backward induction in planning. However, Seidenfeld has demonstrated that such a sophisticated agent with dependent preferences is bound to violate the principle of dynamic substitution, according to which admissibility of a plan is preserved under substitution of indifferent options at various choice nodes in the decision tree. Since Seidenfeld considers dynamic substitution to be a coherence condition on dynamic choice, (...)
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  43. Kenneth Kunen & Dilip Raghavan (2009). Gregory Trees, the Continuum, and Martin's Axiom. Journal of Symbolic Logic 74 (2):712-720.score: 21.0
    We continue the investigation of Gregory trees and the Cantor Tree Property carried out by Hart and Kunen. We produce models of MA with the Continuum arbitrarily large in which there are Gregory trees, and in which there are no Gregory trees.
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  44. Karl‐Heinz Diener (1994). A Remark on Ascending Chain Conditions, the Countable Axiom of Choice and the Principle of Dependent Choices. Mathematical Logic Quarterly 40 (3):415-421.score: 21.0
    It is easy to prove in ZF− that a relation R satisfies the maximal condition if and only if its transitive hull R* does; equivalently: R is well-founded if and only if R* is. We will show in the following that, if the maximal condition is replaced by the chain condition, as is often the case in Algebra, the resulting statement is not provable in ZF− anymore . More precisely, we will prove that this statement is equivalent in ZF− to (...)
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  45. K. Keremedis (2001). The Vector Space Kinna-Wagner Principle is Equivalent to the Axiom of Choice. Mathematical Logic Quarterly 47 (2):205-210.score: 21.0
    We show that the axiom of choice AC is equivalent to the Vector Space Kinna-Wagner Principle, i.e., the assertion: “For every family [MATHEMATICAL SCRIPT CAPITAL V]= {Vi : i ∈ k} of non trivial vector spaces there is a family ℱ = {Fi : i ∈ k} such that for each i ∈ k, Fiis a non empty independent subset of Vi”. We also show that the statement “every vector space over ℚ has a basis” implies that every infinite (...)
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  46. E. Tachtsis & K. Keremedis (2001). Compact Metric Spaces and Weak Forms of the Axiom of Choice. Mathematical Logic Quarterly 47 (1):117-128.score: 21.0
    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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  47. Arthur W. Apter (2011). Indestructibility, HOD, and the Ground Axiom. Mathematical Logic Quarterly 57 (3):261-265.score: 21.0
    Let φ1 stand for the statement V = HOD and φ2 stand for the Ground Axiom. Suppose Ti for i = 1, …, 4 are the theories “ZFC + φ1 + φ2,” “ZFC + ¬φ1 + φ2,” “ZFC + φ1 + ¬φ2,” and “ZFC + ¬φ1 + ¬φ2” respectively. We show that if κ is indestructibly supercompact and λ > κ is inaccessible, then for i = 1, …, 4, Ai = df{δ κ is inaccessible. We show it is (...)
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  48. Norbert Brunner & H. Reiju Mihara (2000). Arrow's Theorem, Weglorz' Models and the Axiom of Choice. Mathematical Logic Quarterly 46 (3):335-359.score: 21.0
    Applying Weglorz' mode s of set theory without the axiom of choice, we investigate Arrow-type social we fare functions for infinite societies with restricted coalition algebras. We show that there is a reasonable, nondictatorial social welfare function satisfying “finite discrimination”, if and only if in Weglorz' mode there is a free ultrafilter on a set representing the individuals.
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  49. Norbert Brunner & Paul Howard (1992). Russell's Alternative to the Axiom of Choice. Mathematical Logic Quarterly 38 (1):529-534.score: 21.0
    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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  50. Norbert Brunner, Karl Svozil & Matthias Baaz (1996). The Axiom of Choice in Quantum Theory. Mathematical Logic Quarterly 42 (1):319-340.score: 21.0
    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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