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  1. Varol Akman (1997). Review of J. Barwise and L. Moss, Vicious Circles: On the Mathematics of Non-Wellfounded Phnenomena. [REVIEW] Journal of Logic, Language and Information 6 (4):460-464.
    This is a review of Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena, by Jon <span class='Hi'>Barwise</span> and Lawrence Moss, published by CSLI (Center for the Study of Language and Information) Publications in 1996.
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  2. P. V. Andreev & E. I. Gordon (2001). An Axiomatics for Nonstandard Set Theory, Based on Von Neumann-Bernays-Gödel Theory. Journal of Symbolic Logic 66 (3):1321-1341.
    We present an axiomatic framework for nonstandard analysis-the Nonstandard Class Theory (NCT) which extends von Neumann-Gödel-Bernays Set Theory (NBG) by adding a unary predicate symbol St to the language of NBG (St(X) means that the class X is standard) and axioms-related to it- analogs of Nelson's idealization, standardization and transfer principles. Those principles are formulated as axioms, rather than axiom schemes, so that NCT is finitely axiomatizable. NCT can be considered as a theory of definable classes of Bounded Set Theory (...)
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  3. Petr Andreev & Karel Hrbacek (2004). Standard Sets in Nonstandard Set Theory. Journal of Symbolic Logic 69 (1):165-182.
    We prove that Standardization fails in every nontrivial universe definable in the nonstandard set theory BST, and that a natural characterization of the standard universe is both consistent with and independent of BST. As a consequence we obtain a formulation of nonstandard class theory in the ∈-language.
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  4. Jonas Rafael Becker Arenhart (2011). A Discussion on Finite Quasi-Cardinals in Quasi-Set Theory. Foundations of Physics 41 (8):1338-1354.
    Quasi-set theory Q is an alternative set-theory designed to deal mathematically with collections of indistinguishable objects. The intended interpretation for those objects is the indistinguishable particles of non-relativistic quantum mechanics, under one specific interpretation of that theory. The notion of cardinal of a collection in Q is treated by the concept of quasi-cardinal, which in the usual formulations of the theory is introduced as a primitive symbol, since the usual means of cardinal definition fail for collections of indistinguishable objects. In (...)
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  5. A. I. Arruda (1982). Russell's set versus the universal set in paraconsistent set theory. Logique Et Analyse 25 (98):121.
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  6. Edward G. Belaga (forthcoming). Retrieving the Mathematical Mission of the Continuum Concept From the Transfinitely Reductionist Debris of Cantor’s Paradise. Extended Abstract. International Journal of Pure and Applied Mathematics.
    What is so special and mysterious about the Continuum, this ancient, always topical, and alongside the concept of integers, most intuitively transparent and omnipresent conceptual and formal medium for mathematical constructions and the battle field of mathematical inquiries ? And why it resists the century long siege by best mathematical minds of all times committed to penetrate once and for all its set-theoretical enigma ? -/- The double-edged purpose of the present study is to save from the transfinite deadlock of (...)
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  7. Maurice Boffa & André Pétry (1993). On Self-Membered Sets in Quine's Set Theory NF. Logique Et Analyse 141:142.
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  8. Justus Diller (2008). Functional Interpretations of Constructive Set Theory in All Finite Types. Dialectica 62 (2):149–177.
    Gödel's dialectica interpretation of Heyting arithmetic HA may be seen as expressing a lack of confidence in our understanding of unbounded quantification. Instead of formally proving an implication with an existential consequent or with a universal antecedent, the dialectica interpretation asks, under suitable conditions, for explicit 'interpreting' instances that make the implication valid. For proofs in constructive set theory CZF-, it may not always be possible to find just one such instance, but it must suffice to explicitly name a set (...)
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  9. Victoria Gitman & Joel David Hamkins (2010). A Natural Model of the Multiverse Axioms. Notre Dame Journal of Formal Logic 51 (4):475-484.
    If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms of Hamkins.
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  10. Luca Incurvati (2014). The Graph Conception of Set. Journal of Philosophical Logic 43 (1):181-208.
    The non-well-founded set theories described by Aczel (1988) have received attention from category theorists and computer scientists, but have been largely ignored by philosophers. At the root of this neglect might lie the impression that these theories do not embody a conception of set, but are rather of mere technical interest. This paper attempts to dispel this impression. I present a conception of set which may be taken as lying behind a non-well-founded set theory. I argue that the axiom AFA (...)
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  11. Ignacio Jané & Gabriel Uzquiano (2004). Well- and Non-Well-Founded Fregean Extensions. Journal of Philosophical Logic 33 (5):437-465.
    George Boolos has described an interpretation of a fragment of ZFC in a consistent second-order theory whose only axiom is a modification of Frege's inconsistent Axiom V. We build on Boolos's interpretation and study the models of a variety of such theories obtained by amending Axiom V in the spirit of a limitation of size principle. After providing a complete structural description of all well-founded models, we turn to the non-well-founded ones. We show how to build models in which foundation (...)
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  12. Christopher Menzel (2014). Wide Sets, ZFCU, and the Iterative Conception. Journal of Philosophy 111 (2):57-83.
    The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence of “wide” sets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as to allow for the (...)
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  13. Richard Pettigrew (2010). The Foundations of Arithmetic in Finite Bounded Zermelo Set Theory. Cahiers du Centre de Logique 17:99-118.
    In this paper, I pursue such a logical foundation for arithmetic in a variant of Zermelo set theory that has axioms of subset separation only for quantifier-free formulae, and according to which all sets are Dedekind finite. In section 2, I describe this variant theory, which I call ZFin0. And in section 3, I sketch foundations for arithmetic in ZFin0 and prove that certain foundational propositions that are theorems of the standard Zermelian foundation for arithmetic are independent of ZFin0.<br><br>An equivalent (...)
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  14. Richard Pettigrew (2009). On Interpretations of Bounded Arithmetic and Bounded Set Theory. Notre Dame Journal of Formal Logic 50 (2):141-152.
    In 'On interpretations of arithmetic and set theory', Kaye and Wong proved the following result, which they considered to belong to the folklore of mathematical logic.

    THEOREM 1 The first-order theories of Peano arithmetic and Zermelo-Fraenkel set theory with the axiom of infinity negated are bi-interpretable.

    In this note, I describe a theory of sets that is bi-interpretable with the theory of bounded arithmetic IDelta0 + exp. Because of the weakness of this theory of sets, I cannot straightforwardly adapt Kaye and Wong's (...)
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  15. Stewart Shapiro & Alan Weir (1999). New V, ZF and Abstraction. Philosophia Mathematica 7 (3):293--321.
    We examine George Boolos's proposed abstraction principle for extensions based on the limitation-of-size conception, New V, from several perspectives. Crispin Wright once suggested that New V could serve as part of a neo-logicist development of real analysis. We show that it fails both of the conservativeness criteria for abstraction principles that Wright proposes. Thus, we support Boolos against Wright. We also show that, when combined with the axioms for Boolos's iterative notion of set, New V yields a system equivalent to (...)
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  16. Hartley Slater (2003). Aggregate Theory Versus Set Theory. Erkenntnis 59 (2):189 - 202.
    Maddy's (1990) arguments against Aggregate Theory were undermined by the shift in her position in 1997. The present paper considers Aggregate Theory in the light of this, and the recent search for `New Axioms for Mathematics'. If Set Theory is the part-whole theory of singletons, then identifying singletons with their single members collapses Set Theory into Aggregate Theory. But if singletons are not identical to their single members, then they are not extensional objects and so are not a basis for (...)
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  17. G. Uzquiano & I. Jané (2004). Well-and Non-Well-Founded Extensions. Journal of Philosophical Logic 33 (5):437-465.
    George Boolos has described an interpretation of a fragment of ZFC in a consistent second-order theory whose only axiom is a modification of Frege's inconsistent Axiom V. We build on Boolos's interpretation and study the models of a variety of such theories obtained by amending Axiom V in the spirit of a limitation of size principle. After providing a complete structural description of all well-founded models, we turn to the non-well-founded ones. We show how to build models in which foundation (...)
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  18. Sylvia Wenmackers & Leon Horsten (2013). Fair Infinite Lotteries. Synthese 190 (1):37-61.
    This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from non-standard analysis are brought to bear on the problem.
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