Results for 'Set theory'

926 found
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  1.  16
    Set Theory.John P. Burgess - 2001 - In Lou Goble, The Blackwell Guide to Philosophical Logic. Malden, Mass.: Wiley-Blackwell. pp. 55–71.
    Set theory is the branch of mathematics concerned with the general properties of aggregates of points, numbers, or arbitrary elements. It was created in the late nineteenth century, mainly by Georg Cantor. After the discovery of certain contradictions euphemistically called paradoxes, it was reduced to axiomatic form in the early twentieth century, mainly by Ernst Zermelo and Abraham Fraenkel. Thereafter it became widely accepted as a framework ‐ or ‘foundation’ ‐ for the development of the other branches of modern, (...)
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  2.  96
    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. The non-triviality of dialectical set theory.Ross T. Brady - 1989 - In Graham Priest, Richard Routley & Jean Norman, Paraconsistent Logic: Essays on the Inconsistent. Philosophia Verlag. pp. 437--470.
     
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  4.  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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  5.  85
    The Reality of Mathematics and the Case of Set Theory.Daniel Isaacson - 2010 - In Zsolt Novák & András Simonyi, Truth, reference, and realism. New York: Central European University Press. pp. 1-76.
  6.  48
    Some properties of intuitionistic Zermelo-Frankel set theory.John Myhill - 1973 - In A. R. D. Mathias & Hartley Rogers, Cambridge Summer School in Mathematical Logic. New York,: Springer Verlag. pp. 206--231.
  7. 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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  8. Arithmetic, Set Theory, Reduction and Explanation.William D’Alessandro - 2018 - Synthese 195 (11):5059-5089.
    Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In (...)
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  9. The Relationship of Arithmetic As Two Twin Peano Arithmetic(s) and Set Theory: A New Glance From the Theory of Information.Vasil Penchev - 2020 - Metaphilosophy eJournal (Elseviers: SSRN) 12 (10):1-33.
    The paper introduces and utilizes a few new concepts: “nonstandard Peano arithmetic”, “complementary Peano arithmetic”, “Hilbert arithmetic”. They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the present such as Fermat’s (...)
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  10.  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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  11.  82
    A set theory with Frege-Russell cardinal numbers.Alan McMichael - 1982 - Philosophical Studies 42 (2):141 - 149.
    A frege-Russell cardinal number is a maximal class of equinumerous classes. Since anything can be numbered, A frege-Russell cardinal should contain classes whose members are cardinal numbers. This is not possible in standard set theories, Since it entails that some classes are members of members of themselves. However, A consistent set theory can be constructed in which such membership circles are allowed and in which, Consequently, Genuine frege-Russell cardinals can be defined.
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  12. 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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  13.  89
    Natural models of Ackermann's set theory.Rudolf Grewe - 1969 - Journal of Symbolic Logic 34 (3):481-488.
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  14. 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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  15.  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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  16.  27
    A study of Maurice fréchet: I. His early work on point set theory and the theory of functionals.Angus E. Taylor - 1982 - Archive for History of Exact Sciences 27 (3):233-295.
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  17.  30
    Finiteness classes arising from Ramsey-theoretic statements in set theory without choice.Joshua Brot, Mengyang Cao & David Fernández-Bretón - 2021 - Annals of Pure and Applied Logic 172 (6):102961.
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  18.  58
    A Dilemma in the Philosophy of Set Theory.Ralf-Dieter Schindler - 1994 - Notre Dame Journal of Formal Logic 35 (3):458-463.
    We show that the following conjecture about the universe V of all sets is wrong: for all set-theoretical (i.e., first order) schemata true in V there is a transitive set "reflecting" in such a way that the second order statement corresponding to is true in . More generally, we indicate the ontological commitments of any theory that exploits reflection principles in order to yield large cardinals. The disappointing conclusion will be that our only apparently good arguments for the existence (...)
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  19.  31
    Lifting proof theory to the countable ordinals: Zermelo-Fraenkel set theory.Toshiyasu Arai - 2014 - Journal of Symbolic Logic 79 (2):325-354.
  20. (1 other version)Modal set theory.Christopher Menzel - 2018 - In Otávio Bueno & Scott A. Shalkowski, The Routledge Handbook of Modality. New York: Routledge.
    This article presents an overview of the basic philosophical motivations for, and some recent work in, modal set theory.
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  21.  31
    Some remarks on Cantor-Lukasiewicz fuzzy set theory.P. Hajek - 2013 - Logic Journal of the IGPL 21 (2):183-186.
  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.  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.
  24. 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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  25.  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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  26.  20
    A set of axioms for the theory of deduction.Bernard Notcutt - 1934 - Mind 43 (169):63-77.
  27.  51
    Set theory generated by Abelian group theory.Paul C. Eklof - 1997 - Bulletin of Symbolic Logic 3 (1):1-16.
    Introduction. This survey is intended to introduce to logicians some notions, methods and theorems in set theory which arose—largely through the work of Saharon Shelah—out of attempts to solve problems in abelian group theory, principally the Whitehead problem and the closely related problem of the existence of almost free abelian groups. While Shelah's first independence result regarding the Whitehead problem used established set-theoretical methods, his later work required new ideas; it is on these that we focus. We emphasize (...)
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  28.  42
    (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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  29.  30
    A set-theoretic model for nonassociative number theory.D. Bollman & M. Laplaza - 1973 - Notre Dame Journal of Formal Logic 14 (1):107-110.
  30.  89
    A Problem in the Foundations of Set Theory.Penelope Maddy - 1990 - Journal of Philosophy 87 (11):619-628.
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  31.  45
    Selected Essays on the History of Set Theory and Logics Philip E. B. Jourdain Ivor Grattan-Guinness.Joan Richards - 1994 - Isis 85 (2):354-355.
  32. Quasi-set theory: a formal approach to a quantum ontology of properties.Federico Holik, Juan Pablo Jorge, Décio Krause & Olimpia Lombardi - 2022 - Synthese 200 (5):1-26.
    In previous works, an ontology of properties for quantum mechanics has been proposed, according to which quantum systems are bundles of properties with no principle of individuality. The aim of the present article is to show that, since quasi-set theory is particularly suited for dealing with aggregates of items that do not belong to the traditional category of individual, it supplies an adequate meta-language to speak of the proposed ontology of properties and its structure.
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  33.  26
    Cavaillès on the Role of Intuition in the Emergence of Set Theory.Daisuke Nakamura - 2013 - Kagaku Tetsugaku 46 (1):53-68.
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  34. Intertwining metaphysics and mathematics: The development of Georg Cantor's set theory 1871-1887.Anne Newstead - 2008 - Review of Contemporary Philosophy 7:35-55.
  35.  36
    Quine W. V.. Unification of universes in set theory.Steven Orey - 1957 - Journal of Symbolic Logic 22 (3):294-295.
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  36.  39
    An extension of Ackermann's set theory.Donald Perlis - 1972 - Journal of Symbolic Logic 37 (4):703-704.
  37.  81
    Cantorian set theory.Alex Oliver & Timothy Smiley - 2018 - Bulletin of Symbolic Logic 24 (4):393-451.
    Almost all set theorists pay at least lip service to Cantor’s definition of a set as a collection of many things into one whole; but empty and singleton sets do not fit with it. Adapting Dana Scott’s axiomatization of the cumulative theory of types, we present a ‘Cantorian’ system which excludes these anomalous sets. We investigate the consequences of their omission, examining their claim to a place on grounds of convenience, and asking whether their absence is an obstacle to (...)
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  38.  43
    Some results about borel sets in descriptive set theory of hyperfinite sets.Boško Živaljević - 1990 - Journal of Symbolic Logic 55 (2):604-614.
  39.  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.
  40.  14
    Logical Foundations of Set Theory and Mathematics.Mary Tiles - 2002 - In Dale Jacquette, A Companion to Philosophical Logic. Malden, MA, USA: Wiley-Blackwell. pp. 365–376.
    This chapter contains sections titled: Foundations and Logical Foundations Foundations for Mathematics Mathematics and Set Theory Sets, Classes, and Logic.
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  41.  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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  42.  64
    On a Collection of Classes in Ackermann's Set Theory With the Axiomschema of Foundation.Klaus Gloede - 1980 - Mathematical Logic Quarterly 26 (32-33):517-522.
  43.  12
    (1 other version)Embedding Properties and Anti‐Foundation in Set Theory.Roland Hinnion - 1989 - Mathematical Logic Quarterly 35 (1):63-70.
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  44.  16
    Ideal topologies in higher descriptive set theory.Peter Holy, Marlene Koelbing, Philipp Schlicht & Wolfgang Wohofsky - 2022 - Annals of Pure and Applied Logic 173 (4):103061.
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  45.  10
    Some Applications of the Theory of Models to Set Theory.H. Jerome Keisler - 1967 - Journal of Symbolic Logic 32 (3):410-410.
  46.  60
    Chandrasekharan K.. A further note on intuitionistic set-theory. The mathematics student, vol. 13 , pp. 49–51.Max Black - 1946 - Journal of Symbolic Logic 11 (4):127-127.
  47.  34
    Normal subgroups of infinite symmetric groups, with an application to stratified set theory.Nathan Bowler & Thomas Forster - 2009 - Journal of Symbolic Logic 74 (1):17-26.
  48.  47
    Robert M. Solovay. A model of set-theory in which every set of reals is Lebesgue measurable. Annals of mathematics, ser. 2 vol. 92 , pp. 1–56. [REVIEW]Richard Laver - 1973 - Journal of Symbolic Logic 38 (3):529.
  49. 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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  50.  72
    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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