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Olivier Esser [13]O. Esser [4]
  1.  16
    On the Consistency of a Positive Theory.Olivier Esser - 1999 - Mathematical Logic Quarterly 45 (1):105-116.
    In positive theories, we have an axiom scheme of comprehension for positive formulas. We study here the “generalized positive” theory GPK∞+. Natural models of this theory are hyperuniverses. The author has shown in [2] that GPK∞+ interprets the Kelley Morse class theory. Here we prove that GPK∞+ + ACWF and the Kelley-Morse class theory with the axiom of global choice and the axiom “On is ramifiable” are mutually interpretable. This shows that GPK∞+ + ACWF is a “strong” theory since “On (...)
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  2.  24
    An Interpretation of the Zermelo‐Fraenkel Set Theory and the Kelley‐Morse Set Theory in a Positive Theory.Olivier Esser - 1997 - Mathematical Logic Quarterly 43 (3):369-377.
    An interesting positive theory is the GPK theory. The models of this theory include all hyperuniverses . Here we add a form of the axiom of infinity and a new scheme to obtain GPK∞+. We show that in these conditions, we can interprete the Kelley-Morse theory in GPK∞+ . This needs a preliminary property which give an interpretation of the Zermelo-Fraenkel set theory in GPK∞+. We also see what happens in the original GPK theory. Before doing this, we first need (...)
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  3.  13
    Inconsistency of GPK + AFA.Olivier Esser - 1996 - Mathematical Logic Quarterly 42 (1):104-108.
    M. Forti and F. Honsell showed in [4] that the hyperuniverses defined in [2] satisfy the anti-foundation axiom X1 introduced in [3]. So it is interesting to study the axiom AFA, which is equivalent to X1 in ZF, introduced by P. Aczel in [1]. We show in this paper that AFA is inconsistent with the theory GPK. This theory, which is first order, is defined by E. Weydert in [6] and later by M. Forti and R. Hinnion in [2]. It (...)
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  4.  36
    Forcing with the Anti‐Foundation Axiom.Olivier Esser - 2012 - Mathematical Logic Quarterly 58 (1-2):55-62.
    In this paper we define the forcing relation and prove its basic properties in the context of the theory ZFCA, i.e., ZFC minus the Foundation axiom and plus the Anti-Foundation axiom.
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  5.  11
    Fixed-Points of Set-Continuous Operators.O. Esser, R. Hinnion & D. Dzierzgowski - 2000 - Mathematical Logic Quarterly 46 (2):183-194.
    In this paper, we study when a set-continuous operator has a fixed-point that is the intersection of a directed family. The framework of our study is the Kelley-Morse theory KMC– and the Gödel-Bernays theory GBC–, both theories including an Axiom of Choice and excluding the Axiom of Foundation. On the one hand, we prove a result concerning monotone operators in KMC– that cannot be proved in GBC–. On the other hand, we study conditions on directed superclasses in GBC– in order (...)
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  6.  32
    On the Axiom of Extensionality in the Positive Set Theory.Olivier Esser - 2003 - Mathematical Logic Quarterly 49 (1):97-100.
    This is a study of the relative interpretability of the axiom of extensionality in the positive set theory. This work has to be considered in the line of works of R. O. Gandy, D. Scott and R. Hinnion who have studied the relative interpretability of the axiom of extensionality in set theories of Zermelo and Zermelo-Fraenkel.
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  7.  19
    Tree‐Properties for Ordered Sets.Olivier Esser & Roland Hinnion - 2002 - Mathematical Logic Quarterly 48 (2):213-219.
    In this paper, we study the notion of arborescent ordered sets, a generalizationof the notion of tree-property for cardinals. This notion was already studied previously in the case of directed sets. Our main result gives a geometric condition for an order to be ℵ0-arborescent.
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  8.  28
    A Strong Model of Paraconsistent Logic.Olivier Esser - 2003 - Notre Dame Journal of Formal Logic 44 (3):149-156.
    The purpose of this paper is mainly to give a model of paraconsistent logic satisfying the "Frege comprehension scheme" in which we can develop standard set theory (and even much more as we shall see). This is the continuation of the work of Hinnion and Libert.
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  9.  12
    On Topological Set Theory.Thierry Libert & Olivier Esser - 2005 - Mathematical Logic Quarterly 51 (3):263-273.
    This paper is concerned with topological set theory, and particularly with Skala's and Manakos' systems for which we give a topological characterization of the models. This enables us to answer natural questions about those theories, reviewing previous results and proving new ones. One of these shows that Skala's set theory is in a sense compatible with any ‘normal’ set theory, and another appears on the semantic side as a ‘Cantor theorem’ for the category of Alexandroff spaces.
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  10.  10
    Tree-Properties for Ordered Sets.Olivier Esser & Roland Hinnion - 2002 - Mathematical Logic Quarterly 48 (2):213-219.
    In this paper, we study the notion of arborescent ordered sets, a generalizationof the notion of tree-property for cardinals. This notion was already studied previously in the case of directed sets. Our main result gives a geometric condition for an order to be ℵ0-arborescent.
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  11.  6
    Topology and Permutations in NF.Olivier Esser - 2007 - Logique Et Analyse 197:87-95.
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  12.  6
    Large Cardinals and Ramifiability for Directed Sets.R. Hinnion & O. Esser - 2000 - Mathematical Logic Quarterly 46 (1):25-34.
    The notion of “ramifiability” , usually applied to cardinals, can be extended to directed sets and is put in relation here with familiar “large cardinal” properties.
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  13.  7
    Antifoundation and Transitive Closure in the System of Zermelo.Olivier Esser & Roland Hinnion - 1999 - Notre Dame Journal of Formal Logic 40 (2):197-205.
    The role of foundation with respect to transitive closure in the Zermelo system Z has been investigated by Boffa; our aim is to explore the role of antifoundation. We start by showing the consistency of "Z antifoundation transitive closure" relative to Z (by a technique well known for ZF). Further, we introduce a "weak replacement principle" (deductible from antifoundation and transitive closure) and study the relations among these three statements in Z via interpretations. Finally, we give some adaptations for ZF (...)
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  14.  4
    Combinatorial Criteria for Ramifiable Ordered Sets.R. Hinnion & O. Esser - 2001 - Mathematical Logic Quarterly 47 (4):539-556.
    The tree-property and its variants make sense also for directed sets and even for partially ordered sets. A combinatoria approach is developed here, with characterizations and criteria involving adequate families of special substructures of directed sets. These substructures form a natural hierarchy that is also investigated.
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  15. A Model of a Strong Paraconsistent Set Theory.O. Esser - 2003 - Notre Dame Journal of Formal Logic 44.