Results for 'Set Theory'

932 found
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  1. Intuitionistic fuzzy logic and intuitionistic fuzzy set theory.Gaisi Takeuti & Satoko Titani - 1984 - Journal of Symbolic Logic 49 (3):851-866.
  2.  94
    Set Theory with Urelements.Bokai Yao - 2023 - Dissertation, University of Notre Dame
    This dissertation aims to provide a comprehensive account of set theory with urelements. In Chapter 1, I present mathematical and philosophical motivations for studying urelement set theory and lay out the necessary technical preliminaries. Chapter 2 is devoted to the axiomatization of urelement set theory, where I introduce a hierarchy of axioms and discuss how ZFC with urelements should be axiomatized. The breakdown of this hierarchy of axioms in the absence of the Axiom of Choice is also (...)
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  3.  71
    Set Theory, Logic and Their Limitations.Moshe Machover - 1996 - Cambridge University Press.
    This is an introduction to set theory and logic that starts completely from scratch. The text is accompanied by many methodological remarks and explanations.
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  4.  72
    The simple consistency of a set theory based on the logic ${\rm CSQ}$.Ross T. Brady - 1983 - Notre Dame Journal of Formal Logic 24 (4):431-449.
  5. The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory.Kurt Gödel - 1940 - Princeton university press;: Princeton University Press;. Edited by George William Brown.
    Kurt Gödel, mathematician and logician, was one of the most influential thinkers of the twentieth century. Gödel fled Nazi Germany, fearing for his Jewish wife and fed up with Nazi interference in the affairs of the mathematics institute at the University of Göttingen. In 1933 he settled at the Institute for Advanced Study in Princeton, where he joined the group of world-famous mathematicians who made up its original faculty. His 1940 book, better known by its short title, The Consistency of (...)
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  6.  84
    A set theory with support for partial functions.William M. Farmer & Joshua D. Guttman - 2000 - Studia Logica 66 (1):59-78.
    Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Gödel set theory for (...)
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  7.  65
    (1 other version)Boolean-Valued Models and Independence Proofs in Set Theory.J. L. Bell & Dana Scott - 1981 - Journal of Symbolic Logic 46 (1):165-165.
  8.  58
    A new model construction by making a detour via intuitionistic theories I: Operational set theory without choice is Π 1 -equivalent to KP.Kentaro Sato & Rico Zumbrunnen - 2015 - Annals of Pure and Applied Logic 166 (2):121-186.
  9.  34
    (1 other version)On strong forms of reflection in set theory.Sy-David Friedman & Radek Honzik - 2016 - Mathematical Logic Quarterly 62 (1-2):52-58.
    In this paper we review the most common forms of reflection and introduce a new form which we call sharp‐generated reflection. We argue that sharp‐generated reflection is the strongest form of reflection which can be regarded as a natural generalization of the Lévy reflection theorem. As an application we formulate the principle sharp‐maximality with the corresponding hypothesis. The statement is an analogue of the (Inner Model Hypothesis, introduced in ) which is compatible with the existence of large cardinals.
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  10. Set Theory and its Philosophy: A Critical Introduction.Michael D. Potter - 2004 - Oxford, England: Oxford University Press.
    Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes (...)
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  11. Set Theory.John P. Burgess - 2022 - Cambridge University Press.
    Set theory is a branch of mathematics with a special subject matter, the infinite, but also a general framework for all modern mathematics, whose notions figure in every branch, pure and applied. This Element will offer a concise introduction, treating the origins of the subject, the basic notion of set, the axioms of set theory and immediate consequences, the set-theoretic reconstruction of mathematics, and the theory of the infinite, touching also on selected topics from higher set (...), controversial axioms and undecided questions, and philosophical issues raised by technical developments. (shrink)
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  12. Set-theoretic absoluteness and the revision theory of truth.Benedikt Löwe & Philip D. Welch - 2001 - Studia Logica 68 (1):21-41.
    We describe the solution of the Limit Rule Problem of Revision Theory and discuss the philosophical consequences of the fact that the truth set of Revision Theory is a complete 1/2 set.
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  13.  36
    Lévy A.. Principles of reflection in axiomatic set theory. Fundamenta mathematicae, vol. 49 no. 1 , pp. 1–10.J. R. Shoenfield - 1965 - Journal of Symbolic Logic 30 (2):251-251.
  14.  53
    Descriptive set theory of families of small sets.Étienne Matheron & Miroslav Zelený - 2007 - Bulletin of Symbolic Logic 13 (4):482-537.
    This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σ-ideals of closed or compact sets.
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  15. Set Theory, Type Theory, and Absolute Generality.Salvatore Florio & Stewart Shapiro - 2014 - Mind 123 (489):157-174.
    In light of the close connection between the ontological hierarchy of set theory and the ideological hierarchy of type theory, Øystein Linnebo and Agustín Rayo have recently offered an argument in favour of the view that the set-theoretic universe is open-ended. In this paper, we argue that, since the connection between the two hierarchies is indeed tight, any philosophical conclusions cut both ways. One should either hold that both the ontological hierarchy and the ideological hierarchy are open-ended, or (...)
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  16.  41
    (2 other versions)Set Theory and its Logic.Willard van Orman Quine - 1963 - Cambridge, MA, USA: Harvard University Press.
    This is an extensively revised edition of Mr. Quine's introduction to abstract set theory and to various axiomatic systematizations of the subject. The treatment of ordinal numbers has been strengthened and much simplified, especially in the theory of transfinite recursions, by adding an axiom and reworking the proofs. Infinite cardinals are treated anew in clearer and fuller terms than before. Improvements have been made all through the book; in various instances a proof has been shortened, a theorem strengthened, (...)
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  17. On self-membered sets in Quine's set theory NF.Maurice Boffa & André Pétry - 1993 - Logique Et Analyse 141:142.
     
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  18. Causal Set Theory and Growing Block? Not Quite.Marco Forgione - manuscript
    In this contribution, I explore the possibility of characterizing the emergence of time in causal set theory (CST) in terms of the growing block universe (GBU) metaphysics. I show that although GBU seems to be the most intuitive time metaphysics for CST, it leaves us with a number of interpretation problems, independently of which dynamics we choose to favor for the theory —here I shall consider the Classical Sequential Growth and the Covariant model. Discrete general covariance of the (...)
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  19.  85
    Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies.Juliette Kennedy & Roman Kossak (eds.) - 2011 - 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 (...)
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  20.  63
    Leslie H. Tharp. On a set theory of Bernays. The journal of symbolic logic, vol. 32 , pp. 319–321.J. R. Shoenfield - 1971 - Journal of Symbolic Logic 36 (4):682.
  21.  51
    Set Theory and Its Logic.J. C. Shepherdson & Willard Van Orman Quine - 1965 - Philosophical Quarterly 15 (61):371.
  22.  40
    32 Naming God’s Essence: Ineffability, Analogy and Set Theory.Claudio Ternullo - 2024 - In Mirosław Szatkowski, Ontology of Divinity. Boston: De Gruyter. pp. 697-718.
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  23.  43
    A study of modal logic with semantics based on rough set theory.Md Aquil Khan, Ranjan & Amal Talukdar - 2024 - Journal of Applied Non-Classical Logics 34 (2):223-247.
    Volume 34, Issue 2-3, June - September 2024.
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  24.  30
    Second order arithmetic as the model companion of set theory.Giorgio Venturi & Matteo Viale - 2023 - Archive for Mathematical Logic 62 (1):29-53.
    This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a Π2\Pi _2 -property formalized in an appropriate language for second order number (...)
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  25.  75
    Finitary Set Theory.Laurence Kirby - 2009 - Notre Dame Journal of Formal Logic 50 (3):227-244.
    I argue for the use of the adjunction operator (adding a single new element to an existing set) as a basis for building a finitary set theory. It allows a simplified axiomatization for the first-order theory of hereditarily finite sets based on an induction schema and a rigorous characterization of the primitive recursive set functions. The latter leads to a primitive recursive presentation of arithmetical operations on finite sets.
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  26. Blunt and topless end extensions of models of set theory.Matt Kaufmann - 1983 - Journal of Symbolic Logic 48 (4):1053-1073.
    Let U be a well-founded model of ZFC whose class of ordinals has uncountable cofinality, such that U has a Σ n end extension for each n ∈ ω. It is shown in Theorem 1.1 that there is such a model which has no elementary end extension. In the process some interesting facts about topless end extensions (those with no least new ordinal) are uncovered, for example Theorem 2.1: If U is a well-founded model of ZFC, such that U has (...)
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  27.  31
    On a positive set theory with inequality.Giacomo Lenzi - 2011 - Mathematical Logic Quarterly 57 (5):474-480.
    We introduce a quite natural Frege-style set theory, which we call Strong-Frege-2 equation image, a sort of simplification of the theory considered in 13 and 1 . We give a model of a weaker variant of equation image, called equation image, where atoms and coatoms are allowed. To construct the model we use an enumeration “almost without repetitions” of the Π11 sets of natural numbers; such an enumeration can be obtained via a classical priority argument much in the (...)
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  28.  16
    On Models of Zermelo-Fraenkel Set Theory Satisfying the Axiom of Constructibility.Andrzej Mostowski - 1971 - Journal of Symbolic Logic 36 (3):542-542.
  29.  27
    Set Theory: Boolean-Valued Models and Independence Proofs.John L. Bell - 2011 - Oxford University Press.
    This third edition, now available in paperback, is a follow up to the author's classic Boolean-Valued Models and Independence Proofs in Set Theory. It provides an exposition of some of the most important results in set theory obtained in the 20th century: the independence of the continuum hypothesis and the axiom of choice.
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  30.  47
    Nested sets theory, full stop: Explaining performance on bayesian inference tasks without dual-systems assumptions.David R. Mandel - 2007 - Behavioral and Brain Sciences 30 (3):275-276.
    Consistent with Barbey & Sloman (B&S), it is proposed that performance on Bayesian inference tasks is well explained by nested sets theory (NST). However, contrary to those authors' view, it is proposed that NST does better by dispelling with dual-systems assumptions. This article examines why, and sketches out a series of NST's core principles, which were not previously defined.
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  31.  66
    A constructive interpretation of the full set theory.Valentin F. Turchin - 1987 - Journal of Symbolic Logic 52 (1):172-201.
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  32. Foundations of Set Theory.Abraham Adolf Fraenkel & Yehoshua Bar-Hillel - 1973 - Atlantic Highlands, NJ, USA: Elsevier.
    Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a (...)
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  33.  49
    Arithmetical set theory.Paul Strauss - 1991 - Studia Logica 50 (2):343 - 350.
    It is well known that number theory can be interpreted in the usual set theories, e.g. ZF, NF and their extensions. The problem I posed for myself was to see if, conversely, a reasonably strong set theory could be interpreted in number theory. The reason I am interested in this problem is, simply, that number theory is more basic or more concrete than set theory, and hence a more concrete foundation for mathematics. A partial solution (...)
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  34.  37
    Quantum set theory: Transfer Principle and De Morgan's Laws.Masanao Ozawa - 2021 - Annals of Pure and Applied Logic 172 (4):102938.
    In quantum logic, introduced by Birkhoff and von Neumann, De Morgan's Laws play an important role in the projection-valued truth value assignment of observational propositions in quantum mechanics. Takeuti's quantum set theory extends this assignment to all the set-theoretical statements on the universe of quantum sets. However, Takeuti's quantum set theory has a problem in that De Morgan's Laws do not hold between universal and existential bounded quantifiers. Here, we solve this problem by introducing a new truth value (...)
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  35. The real line in elementary submodels of set theory.Kenneth Kunen & Franklin Tall - 2000 - Journal of Symbolic Logic 65 (2):683-691.
    Keywords: Elementary Submodel; Real Line; Order-Isomorphic.
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  36.  17
    Simplified Independence Proofs. Boolean Valued Models of Set Theory.J. Barkley Rosser - 1974 - Journal of Symbolic Logic 39 (2):328-329.
  37.  14
    (1 other version)Elementary Equivalence and Constructible Models of Zermelo‐Fraenkel Set Theory.R. H. Cowen - 1976 - Mathematical Logic Quarterly 22 (1):333-338.
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  38. On the standard part of nonstandard models of set theory.Menachem Magidor, Saharon Shelah & Jonathan Stavi - 1983 - Journal of Symbolic Logic 48 (1):33-38.
    We characterize the ordinals α of uncountable cofinality such that α is the standard part of a nonstandard model of ZFC (or equivalently KP).
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  39.  13
    An (α,β)-Hesitant Fuzzy Set Approach to Ideal Theory in Semigroups.Pairote Yiarayong - 2022 - Bulletin of the Section of Logic 51 (3):383-409.
    The aim of this manuscript is to introduce the (α,β)(\alpha,\beta)-hesitant fuzzy set and apply it to semigroups. In this paper, as a generalization of the concept of hesitant fuzzy sets to semigroup theory, the concept of (α,β)(\alpha,\beta)-hesitant fuzzy subsemigroups of semigroups is introduced, and related properties are discussed. Furthermore, we define and study (α,β)(\alpha,\beta)-hesitant fuzzy ideals on semigroups. In particular, we investigate the structure of (α,β)(\alpha,\beta)-hesitant fuzzy ideal generated by a hesitant fuzzy ideal in a semigroup. In addition, we (...)
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  40.  32
    Algebraic Set Theory and the Effective Topos.Claire Kouwenhoven-Gentil & Jaap van Oosten - 2005 - Journal of Symbolic Logic 70 (3):879 - 890.
    Following the book Algebraic Set Theory from André Joyal and leke Moerdijk [8], we give a characterization of the initial ZF-algebra, for Heyting pretoposes equipped with a class of small maps. Then, an application is considered (the effective topos) to show how to recover an already known model (McCarty [9]).
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  41.  13
    Functional Interpretations of Classical and Constructive Set Theory.Justus Diller - 2012 - In Ulrich Berger, Hannes Diener, Peter Schuster & Monika Seisenberger, Logic, Construction, Computation. De Gruyter. pp. 137-156.
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  42. A model of a strong paraconsistent set theory.O. Esser - 2003 - Notre Dame Journal of Formal Logic 44.
  43.  85
    A long-awaited edition of Zermelo’s works: Ernst Zermelo: Collected works/gesammelte Werke. Vol. I: Set theory, miscellanea. Edited by H.-D. Ebbinghaus and A. Kanamori. Berlin: Springer, 2010, xxiv+654pp, €109.95 HB.José Ferreirós - 2011 - Metascience 20 (3):505-508.
    A long-awaited edition of Zermelo’s works Content Type Journal Article Pages 1-4 DOI 10.1007/s11016-011-9548-y Authors José Ferreirós, Instituto de Filosofia, CCHS-CSIC, Albasanz, 26-28, 28037 Madrid, Spain Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.
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  44.  66
    Cantorian set Theory and Limitation of Size.John Mayberry - 1986 - Philosophical Quarterly 36 (144):429-434.
    This is a book review of Cantorian set theory and limitations of size by Michael Hallett.
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  45.  19
    The Interpretation of Classes in Axiomatic Set Theory.Gregor Schneider & Daniel Roth - 2014 - In Godehard Link, Formalism and Beyond: On the Nature of Mathematical Discourse. Boston: De Gruyter. pp. 275-314.
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  46.  79
    Finite Cardinals in Quasi-set Theory.Jonas R. Becker Arenhart - 2012 - Studia Logica 100 (3):437-452.
    Quasi-set theory is a ZFU-like axiomatic set theory, which deals with two kinds of ur-elements: M-atoms, objects like the atoms of ZFU, and m-atoms, items for which the usual identity relation is not defined. One of the motivations to advance such a theory is to deal properly with collections of items like particles in non-relativistic quantum mechanics when these are understood as being non-individuals in the sense that they may be indistinguishable although identity does not apply to (...)
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  47.  66
    A Version of Kripke‐Platek Set Theory Which is Conservative Over Peano Arithmetic.Gerhard Jäger - 1984 - Mathematical Logic Quarterly 30 (1-6):3-9.
  48.  22
    Choice and independence of premise rules in intuitionistic set theory.Emanuele Frittaion, Takako Nemoto & Michael Rathjen - 2023 - Annals of Pure and Applied Logic 174 (9):103314.
  49.  15
    Logical and Gnoseology Analysis of Fuzzy Set Theory of Lotfi Zadeh.Fuad Gurbanov & Aziz Mammadov - 2019 - Metafizika 2 (1):7-29.
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  50.  13
    (1 other version)Correction to “Embedding Properties and Anti‐Foundation in Set Theory”.Roland Hinnion - 1989 - Mathematical Logic Quarterly 35 (6):574-574.
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