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  1. F. G. Asenjo (1977). Leśniewski's Work and Nonclassical Set Theories. Studia Logica 36 (4):249-255.
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  2. Zdzisław Augustynek (1994). Z ontologii czasoprzestrzeni. Filozofia Nauki 2.
    The question concerning the ontic nature of space-time points and of space-time itself - is the question: are these objects set-theoretic sets or individuals, i.e. nonsets? Two classifications of the standpoints concerning the nature of these objects are formulated and then they are intersected. In concequence three standpoints appear: mereological substantivalism, set-theoretic substantivalism and set-theoretical relationism; it is showed that mereological relationism is not real. It is proved that set-theoretic standpoints logically imply so called set-theoretic realism which accepts the existence (...)
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  3. John L. Bell (2000). Sets and Classes as Many. Journal of Philosophical Logic 29 (6):585-601.
    In this paper the view is developed that classes should not be understood as individuals, but, rather, as "classes as many" of individuals. To correlate classes with individuals "labelling" and "colabelling" functions are introduced and sets identified with a certain subdomain of the classes on which the labelling and colabelling functions are mutually inverse. A minimal axiomatization of the resulting system is formulated and some of its extensions are related to various systems of set theory, including nonwellfounded set theories.
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  4. Andreas Blass & Andre Scedrov (1992). Complete Topoi Representing Models of Set Theory. Annals of Pure and Applied Logic 57 (1):1-26.
    By a model of set theory we mean a Boolean-valued model of Zermelo-Fraenkel set theory allowing atoms (ZFA), which contains a copy of the ordinary universe of (two-valued,pure) sets as a transitive subclass; examples include Scott-Solovay Boolean-valued models and their symmetric submodels, as well as Fraenkel-Mostowski permutation models. Any such model M can be regarded as a topos. A logical subtopos E of M is said to represent M if it is complete and its cumulative hierarchy, as defined by Fourman (...)
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  5. Manuel Bremer (2010). Universality in Set Theories. Ontos.
    The book discusses the fate of universality and a universal set in several set theories.
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  6. Ray-Ming Chen & Michael Rathjen (2012). Lifschitz Realizability for Intuitionistic Zermelo–Fraenkel Set Theory. Archive for Mathematical Logic 51 (7-8):789-818.
    A variant of realizability for Heyting arithmetic which validates Church’s thesis with uniqueness condition, but not the general form of Church’s thesis, was introduced by Lifschitz (Proc Am Math Soc 73:101–106, 1979). A Lifschitz counterpart to Kleene’s realizability for functions (in Baire space) was developed by van Oosten (J Symb Log 55:805–821, 1990). In that paper he also extended Lifschitz’ realizability to second order arithmetic. The objective here is to extend it to full intuitionistic Zermelo–Fraenkel set theory, IZF. The machinery (...)
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  7. Peter Clark (1998). Dummett's Argument for the Indefinite Extensibility of Set and Real Number. Grazer Philosophische Studien 55:51-63.
    The paper examines Dummett's argument for the indefinite extensibility of the concepts set, ordinal, real number, set of natural numbers, and natural number. In particular it investigates how the indefinite extensibility of the concept set affects our understanding of the notion of real number and whether the argument to the indefinite extensibility of the reals is cogent. It claims that Dummett is right to think of the universe of sets as an indefinitely extensible domain but questions the cogency of the (...)
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  8. Clinton T. Conley (2013). Canonizing Relations on Nonsmooth Sets. Journal of Symbolic Logic 78 (1):101-112.
    We show that any symmetric, Baire measurable function from the complement of $\ezero$ to a finite set is constant on an $\ezero$-nonsmooth square. A simultaneous generalization of Galvin's theorem that Baire measurable colorings admit perfect homogeneous sets and the Kanovei-Zapletal theorem canonizing Borel equivalence relations on $E_0$-nonsmooth sets, this result is proved by relating $\ezero$-nonsmooth sets to embeddings of the complete binary tree into itself and appealing to a version of Hindman's theorem on the complete binary tree. We also establish (...)
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  9. Adam R. Day (2013). Indifferent Sets for Genericity. Journal of Symbolic Logic 78 (1):113-138.
    This paper investigates indifferent sets for comeager classes in Cantor space focusing of the class of all 1-generic sets and the class of all weakly 1-generic sets. Jockusch and Posner showed that there exist 1-generic sets that have indifferent sets [10]. Figueira, Miller and Nies have studied indifferent sets for randomness and other notions [7]. We show that any comeager class in Cantor space contains a comeager class with a universal indifferent set. A forcing construction is used to show that (...)
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  10. J. Ferreiros (2004). The Motives Behind Cantor’s Set Theory: Physical, Biological and Philosophical Questions. Science in Context 17 (1/2):1–35.
    The celebrated “creation” of transfinite set theory by Georg Cantor has been studied in detail by historians of mathematics. However, it has generally been overlooked that his research program cannot be adequately explained as an outgrowth of the mainstream mathematics of his day. We review the main extra-mathematical motivations behind Cantor's very novel research, giving particular attention to a key contribution, the Grundlagen (Foundations of a general theory of sets) of 1883, where those motives are articulated in some detail. Evidence (...)
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  11. Joseph S. Fulda (2009). Rendering Conditionals in Mathematical Discourse with Conditional Elements. Journal of Pragmatics 41 (7):1435-1439.
    In "Material Implications" (1992), mathematical discourse was said to be different from ordinary discourse, with the discussion centering around conditionals. This paper shows how.
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  12. Klaus Gloede (1977). The Metamathematics of Infinitary Set Theoretical Systems. Mathematical Logic Quarterly 23 (1‐6):19-44.
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  13. Joel David Hamkins, David Linetsky & Jonas Reitz (2013). Pointwise Definable Models of Set Theory. Journal of Symbolic Logic 78 (1):139-156.
    A pointwise definable model is one in which every object is \loos definable without parameters. In a model of set theory, this property strengthens $V=\HOD$, but is not first-order expressible. Nevertheless, if \ZFC\ is consistent, then there are continuum many pointwise definable models of \ZFC. If there is a transitive model of \ZFC, then there are continuum many pointwise definable transitive models of \ZFC. What is more, every countable model of \ZFC\ has a class forcing extension that is pointwise definable. (...)
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  14. A. P. Hazen (1993). Against Pluralism. Australasian Journal of Philosophy 71 (2):132 – 144.
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  15. Luca Incurvati (2012). How to Be a Minimalist About Sets. Philosophical Studies 159 (1):69-87.
    According to the iterative conception of set, sets can be arranged in a cumulative hierarchy divided into levels. But why should we think this to be the case? The standard answer in the philosophical literature is that sets are somehow constituted by their members. In the first part of the paper, I present a number of problems for this answer, paying special attention to the view that sets are metaphysically dependent upon their members. In the second part of the paper, (...)
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  16. Luca Incurvati & Julien Murzi (forthcoming). Maximally Consistent Sets of Instances of Naive Comprehension. Mind.
    Paul Horwich (1990) once suggested restricting the T-Schema to the maximal consistent set of its instances. But Vann McGee (1992) proved that there are multiple incompatible such sets, none of which, given minimal assumptions, is recursively axiomatizable. The analogous view for set theory---that Naïve Comprehension should be restricted according to consistency maxims---has recently been defended by Laurence Goldstein (2006; 2013). It can be traced back to W.V.O. Quine(1951), who held that Naïve Comprehension embodies the only really intuitive conception of set (...)
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  17. Juliette Kennedy & Roman Kossak (eds.) (2012). Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies. Cambridge University Press.
    Machine generated contents note: 1. Introduction Juliette Kennedy and Roman Kossak; 2. Historical remarks on Suslin's problem Akihiro Kanamori; 3. The continuum hypothesis, the generic-multiverse of sets, and the [OMEGA] conjecture W. Hugh Woodin; 4. [omega]-Models of finite set theory Ali Enayat, James H. Schmerl and Albert Visser; 5. Tennenbaum's theorem for models of arithmetic Richard Kaye; 6. Hierarchies of subsystems of weak arithmetic Shahram Mohsenipour; 7. Diophantine correct open induction Sidney Raffer; 8. Tennenbaum's theorem and recursive reducts James H. (...)
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  18. Øystein Linnebo (2003). Plural Quantification Exposed. Noûs 37 (1):71–92.
  19. Øystein Linnebo & Richard Pettigrew (2011). Category Theory as an Autonomous Foundation. Philosophia Mathematica 19 (3):227-254.
    Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in other (...)
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  20. Jean-Pierre Marquis (2006). Categories, Sets and the Nature of Mathematical Entities. In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. 181--192.
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  21. Thomas J. McKay (2006). Plural Predication. Oxford University Press.
    Plural predication is a pervasive part of ordinary language. We can say that some people are fifty in number, are surrounding a building, come from many countries, and are classmates. These predicates can be true of some people without being true of any one of them; they are non-distributive predications. However, the apparatus of modern logic does not allow a place for them. Thomas McKay here explores the enrichment of logic with non-distributive plural predication and quantification. His book will be (...)
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  22. Daniel Nolan, Individuals Enough for Classes.
    This paper builds on the system of David Lewis’s “Parts of Classes” to provide a foundation for mathematics that arguably requires not only no distinctively mathematical ideological commitments (in the sense of Quine), but also no distinctively mathematical ontological commitments. Provided only that there are enough individual atoms, the devices of plural quantification and mereology can be employed to simulate quantification over classes, while at the same time allowing all of the atoms (and most of their fusions with which we (...)
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  23. Chang Kyun Park (2008). A Philosophical Interpretation of Rough Set Theory. Proceedings of the Xxii World Congress of Philosophy 13:23-29.
    The rough set theory has interesting properties such as that a rough set is considered as distinct sets in distinct knowledge bases, and that distinct rough sets are considered as one same set in a certain knowledge base. This leads to a significant philosophical interpretation: a concept (or phenomenon) may be understood as different ones in different philosophical perspectives, while different concepts (or phenomena) may be understood as a same one in a certain philosophical perspective. Such properties of rough set (...)
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  24. Matthew W. Parker (2003). Three Concepts of Decidability for General Subsets of Uncountable Spaces. Theoretical Computer Science 351 (1):2-13.
    There is no uniquely standard concept of an effectively decidable set of real numbers or real n-tuples. Here we consider three notions: decidability up to measure zero [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382], which we abbreviate d.m.z.; recursive approximability [or r.a.; K.-I. Ko, Complexity Theory of Real Functions, Birkhäuser, Boston, 1991]; and decidability ignoring boundaries [d.i.b.; W.C. Myrvold, The decision problem for entanglement, in: R.S. (...)
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  25. Nicholas Rescher & Patrick Grim (2008). Plenum Theory. Noûs 42 (3):422-439.
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  26. Michael J. Shaffer (2006). Some Recent Existential Appeals to Mathematical Experience. Principia 10 (2):143-170.
    Some recent work by philosophers of mathematics has been aimed at showing that our knowledge of the existence of at least some mathematical objects and/or sets can be epistemically grounded by appealing to perceptual experience. The sensory capacity that they refer to in doing so is the ability to perceive numbers, mathematical properties and/or sets. The chief defense of this view as it applies to the perception of sets is found in Penelope Maddy’s Realism in Mathematics, but a number of (...)
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  27. Stewart Shapiro (2003). All Sets Great and Small: And I Do Mean ALL. Philosophical Perspectives 17 (1):467–490.
    A number of authors have recently weighed in on the issue of whether it is coherent to have bound variables that range over absolutely everything. Prima facie, it is difficult, and perhaps impossible, to coherently state the “relativist” position without violating it. For example, the relativist might say, or try to say, that for any quantifier used in a proposition of English, there is something outside of its range. What is the range of this quantifier? Or suppose we ask the (...)
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  28. Antonin Sochor (1975). Contribution to the Theory of Semisets VI: (Non‐Existence of the Class of All Absolute Natural Numbers). Mathematical Logic Quarterly 21 (1):439-442.
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  29. Mary Tiles (2008). Review of T. Arrigoni, What is Meant by V?: Reflections on the Universe of All Sets. [REVIEW] Philosophia Mathematica 16 (1):132-133.
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  30. Gabriel Uzquiano (forthcoming). Varieties of Indefinite Extensibility. Notre Dame Journal of Formal Logic.
    We look at two recent accounts of the indefinite extensibility of set, and compare them with a linguistic model of the indefinite extensibility. I suggest the linguistic model has much to recommend over extant accounts of the indefinite extensibility of set, and we defend it against three prima facie objections.
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  31. Gabriel Uzquiano (2002). Categoricity Theorems and Conceptions of Set. Journal of Philosophical Logic 31 (2):181-196.
    Two models of second-order ZFC need not be isomorphic to each other, but at least one is isomorphic to an initial segment of the other. The situation is subtler for impure set theory, but Vann McGee has recently proved a categoricity result for second-order ZFCU plus the axiom that the urelements form a set. Two models of this theory with the same universe of discourse need not be isomorphic to each other, but the pure sets of one are isomorphic to (...)
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  32. Hao Wang (1959). Ordinal Numbers and Predicative Set Theory. Mathematical Logic Quarterly 5 (14‐24):216-239.
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  33. Jan Westerhoff (2004). A Taxonomy of Composition Operations. Logique and Analyse 2004 (47):375-393.
    A set of parameters for classifying composition operations is introduced. These parameters determine whether a composition operation is 1) universal, 2) determinate, 3) whether there is a difference between possible and actual compositions, 4) whether there can be singleton compositions, 5) whether they give rise to a hierarchy, and 6) whether components of compositions can be repeated. Philosophical implications of these parameters (in particular in relation to set theory) and mereology are discussed.
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  34. Piotr Wilczek (2010). Nowy postulat teorii mnogości – aksjomat Leibniza-Mycielskiego. Filozofia Nauki 3.
    In this article we will present the Leibniz-Mycielski axiom (LM) of set theory (ZF) introduced several years ago by Jan Mycielski as an additional axiom of set theory. This new postulate formalizes the so-called Leibniz Law (LL) which states that there are no two distinct indiscernible objects. From the Ehrenfeucht-Mostowski theorem it follows that every theory which has an infinite model has a model with indiscernibles. The new LM axiom states that there are infinite models without indis-cernibles. These models are (...)
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