Results for 'set theory'

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  1.  53
    Set Theory and its Logic: Revised Edition.Willard Van Orman Quine - 1963 - 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.
  2. 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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  3. Set theory and the continuum hypothesis.Paul J. Cohen - 1966 - New York,: W. A. Benjamin.
    This exploration of a notorious mathematical problem is the work of the man who discovered the solution. Written by an award-winning professor at Stanford University, it employs intuitive explanations as well as detailed mathematical proofs in a self-contained treatment. This unique text and reference is suitable for students and professionals. 1966 edition. Copyright renewed 1994.
  4. 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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  5.  55
    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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  6.  4
    Set Theory : Boolean-Valued Models and Independence Proofs: Boolean-Valued Models and Independence Proofs.John L. Bell - 2005 - Oxford University Press UK.
    This monograph is a follow up to the author's classic text Boolean-Valued Models and Independence Proofs in Set Theory, providing 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. Aimed at research students and academics in mathematics, mathematical logic, philosophy, and computer science, the text has been extensively updated with expanded introductory material, new chapters, and a new appendix on category (...)
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  7. 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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  8. Wand/set theories.Tim Button - manuscript
    Here is a template for introducing mathematical objects: “Objects are found in stages. For every stage S: (1) for any things found before S, you find at S the bland set whose members are exactly those things; (2) for anything, x, which was found before S, you find at S the result of tapping x with any magic wand (provided that the result is not itself a bland set); you find nothing else at S.” -/- This Template has rich applications, (...)
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  9.  14
    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, (...)
  10.  32
    Set Theory and Its Logic.J. C. Shepherdson & Willard Van Orman Quine - 1965 - Philosophical Quarterly 15 (61):371.
  11.  19
    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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  12. Modal set theory.Christopher Menzel - 2018 - In Otávio Bueno & Scott A. Shalkowski (eds.), The Routledge Handbook of Modality. London and 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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  13.  79
    Naïve set theory is innocent!A. Weir - 1998 - Mind 107 (428):763-798.
    Naive set theory, as found in Frege and Russell, is almost universally believed to have been shown to be false by the set-theoretic paradoxes. The standard response has been to rank sets into one or other hierarchy. However it is extremely difficult to characterise the nature of any such hierarchy without falling into antinomies as severe as the set-theoretic paradoxes themselves. Various attempts to surmount this problem are examined and criticised. It is argued that the rejection of naive set (...)
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  14. 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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  15.  15
    Set Theory. An Introduction to Independence Proofs.James E. Baumgartner & Kenneth Kunen - 1986 - Journal of Symbolic Logic 51 (2):462.
  16.  43
    Naive Set Theory and Nontransitive Logic.David Ripley - 2015 - Review of Symbolic Logic 8 (3):553-571.
    In a recent series of papers, I and others have advanced new logical approaches to familiar paradoxes. The key to these approaches is to accept full classical logic, and to accept the principles that cause paradox, while preventing trouble by allowing a certain sort ofnontransitivity. Earlier papers have treated paradoxes of truth and vagueness. The present paper will begin to extend the approach to deal with the familiar paradoxes arising in naive set theory, pointing out some of the promises (...)
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  17.  16
    Constructive Set Theory with Operations.Andrea Cantini & Laura Crosilla - 2008 - In Logic Colloquium 2004.
    We present an extension of constructive Zermelo{Fraenkel set theory [2]. Constructive sets are endowed with an applicative structure, which allows us to express several set theoretic constructs uniformly and explicitly. From the proof theoretic point of view, the addition is shown to be conservative. In particular, we single out a theory of constructive sets with operations which has the same strength as Peano arithmetic.
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  18.  34
    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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  19.  9
    Set Theory.Keith J. Devlin - 1981 - Journal of Symbolic Logic 46 (4):876-877.
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  20.  30
    Cantorian Set Theory and Limitation of Size.Michael Hallett - 1984 - Oxford, England: Clarendon Press.
    This volume presents the philosophical and heuristic framework Cantor developed and explores its lasting effect on modern mathematics. "Establishes a new plateau for historical comprehension of Cantor's monumental contribution to mathematics." --The American Mathematical Monthly.
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  21. Constructive set theory.John Myhill - 1975 - Journal of Symbolic Logic 40 (3):347-382.
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  22.  35
    Set theory.Thomas Jech - 1981 - Journal of Symbolic Logic.
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  23. Set Theory.T. Jech - 2005 - Bulletin of Symbolic Logic 11 (2):243-245.
     
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  24.  10
    Lectures in set theory.Thomas J. Jech - 1971 - New York,: Springer Verlag.
  25.  69
    Operational set theory and small large cardinals.Solomon Feferman with with R. L. Vaught - manuscript
    “Small” large cardinal notions in the language of ZFC are those large cardinal notions that are consistent with V = L. Besides their original formulation in classical set theory, we have a variety of analogue notions in systems of admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics and recursive ordinal notations (as used in proof theory). On the face of it, it is surprising that such distinctively set-theoretical notions (...)
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  26.  43
    Finitist set theory in ontological modeling.Avril Styrman & Aapo Halko - 2018 - Applied ontology 13 (2):107-133.
    This article introduces finitist set theory (FST) and shows how it can be applied in modeling finite nested structures. Mereology is a straightforward foundation for transitive chains of part-whole relations between individuals but is incapable of modeling antitransitive chains. Traditional set theories are capable of modeling transitive and antitransitive chains of relations, but due to their function as foundations of mathematics they come with features that make them unnecessarily difficult in modeling finite structures. FST has been designed to function (...)
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  27.  57
    Nonstandard set theory.Peter Fletcher - 1989 - Journal of Symbolic Logic 54 (3):1000-1008.
    Nonstandard set theory is an attempt to generalise nonstandard analysis to cover the whole of classical mathematics. Existing versions (Nelson, Hrbáček, Kawai) are unsatisfactory in that the unlimited idealisation principle conflicts with the wish to have a full theory of external sets. I re-analyse the underlying requirements of nonstandard set theory and give a new formal system, stratified nonstandard set theory, which seems to meet them better than the other versions.
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  28.  27
    Set theory and the continuum problem.Raymond Smullyan - 1996 - Clarendon Press.
    A lucid, elegant, and complete survey of set theory, this three-part treatment explores axiomatic set theory, the consistency of the continuum hypothesis, and forcing and independence results. 1996 edition.
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  29.  43
    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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  30. Cantorian Set Theory and Limitation of Size.Michael Hallett - 1990 - Studia Logica 49 (2):283-284.
     
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  31.  23
    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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  32. Does set theory really ground arithmetic truth?Alfredo Roque Freire - manuscript
    We consider the foundational relation between arithmetic and set theory. Our goal is to criticize the construction of standard arithmetic models as providing grounds for arithmetic truth (even in a relative sense). Our method is to emphasize the incomplete picture of both theories and treat models as their syntactical counterparts. Insisting on the incomplete picture will allow us to argue in favor of the revisability of the standard model interpretation. We then show that it is hopeless to expect that (...)
     
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  33.  14
    Descriptive Set Theory.Richard Mansfield - 1981 - Journal of Symbolic Logic 46 (4):874-876.
  34.  56
    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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  35.  13
    Introduction to axiomatic set theory.Gaisi Takeuti - 1971 - New York,: Springer Verlag. Edited by Wilson M. Zaring.
    In 1963, the first author introduced a course in set theory at the University of Illinois whose main objectives were to cover Godel's work on the con sistency of the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH), and Cohen's work on the independence of the AC and the GCH. Notes taken in 1963 by the second author were taught by him in 1966, revised extensively, and are presented here as an introduction to axiomatic set theory. (...)
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  36. Cantorian Set Theory and Limitation of Size.Michael Hallett - 1986 - Mind 95 (380):523-528.
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  37.  56
    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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  38.  13
    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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  39. Multiverse Conceptions in Set Theory.Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo - 2015 - Synthese 192 (8):2463-2488.
    We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...)
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  40.  20
    Naive Set Theory with Extensionality in Partial Logic and in Paradoxical Logic.Roland Hinnion - 1994 - Notre Dame Journal of Formal Logic 35 (1):15-40.
    Two distinct and apparently "dual" traditions of non-classical logic, three-valued logic and paraconsistent logic, are considered here and a unified presentation of "easy-to-handle" versions of these logics is given, in which full naive set theory, i.e. Frege's comprehension principle + extensionality, is not absurd.
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  41. Set Theory: An Introduction to Large Cardinals.F. R. Drake & T. J. Jech - 1976 - British Journal for the Philosophy of Science 27 (2):187-191.
     
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  42.  31
    Set Theory and Its Logic.Joseph S. Ullian & Willard Van Orman Quine - 1966 - Philosophical Review 75 (3):383.
  43. Set Theory, Topology, and the Possibility of Junky Worlds.Thomas Mormann - 2014 - Notre Dame Journal of Formal Logic 55 (1): 79 - 90.
    A possible world is a junky world if and only if each thing in it is a proper part. The possibility of junky worlds contradicts the principle of general fusion. Bohn (2009) argues for the possibility of junky worlds, Watson (2010) suggests that Bohn‘s arguments are flawed. This paper shows that the arguments of both authors leave much to be desired. First, relying on the classical results of Cantor, Zermelo, Fraenkel, and von Neumann, this paper proves the possibility of junky (...)
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  44.  17
    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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  45. Nonstandard set theories and information management.Varol Akman & Mujdat Pakkan - 1996 - Journal of Intelligent Information Systems 6:5-31.
    The merits of set theory as a foundational tool in mathematics stimulate its use in various areas of artificial intelligence, in particular intelligent information systems. In this paper, a study of various nonstandard treatments of set theory from this perspective is offered. Applications of these alternative set theories to information or knowledge management are surveyed.
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  46.  23
    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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  47.  40
    Set theory, model theory, and computability theory.Wilfrid Hodges - 2009 - In Leila Haaparanta (ed.), The Development of Modern Logic. Oxford University Press. pp. 471.
    This chapter surveys set theory, model theory, and computability theory: how they first emerged from the foundations of mathematics, and how they have developed since. There are any amounts of mathematical technicalities in the background, but the chapter highlights those themes that have some philosophical resonance.
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  48. Naive Set Theory.Paul R. Halmos & Patrick Suppes - 1961 - Synthese 13 (1):86-87.
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  49.  50
    Set theory: Constructive and intuitionistic ZF.Laura Crosilla - 2010 - Stanford Encyclopedia of Philosophy.
    Constructive and intuitionistic Zermelo-Fraenkel set theories are axiomatic theories of sets in the style of Zermelo-Fraenkel set theory (ZF) which are based on intuitionistic logic. They were introduced in the 1970's and they represent a formal context within which to codify mathematics based on intuitionistic logic. They are formulated on the basis of the standard first order language of Zermelo-Fraenkel set theory and make no direct use of inherently constructive ideas. In working in constructive and intuitionistic ZF we (...)
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  50.  39
    Set Theory. An Introduction to Large Cardinals.Azriel Levy - 1978 - Journal of Symbolic Logic 43 (2):384-384.
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