Results for 'set theory'

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  1.  7
    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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  2.  48
    Set Theory and its Logic: Revised Edition.Willard V. 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.
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  3. 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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  4.  70
    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.
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  5. 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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  6.  51
    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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  7. Foundations of Set Theory.Abraham Adolf Fraenkel & Yehoshua Bar-Hillel - 1958 - Atlantic Highlands, NJ, USA: North-Holland.
    HISTORICAL INTRODUCTION In Abstract Set Theory) the elements of the theory of sets were presented in a chiefly generic way: the fundamental concepts were ...
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  8.  78
    Kripke-Platek Set Theory and the Anti-Foundation Axiom.Michael Rathjen - 2001 - Mathematical Logic Quarterly 47 (4):435-440.
    The paper investigates the strength of the Anti-Foundation Axiom, AFA, on the basis of Kripke-Platek set theory without Foundation. It is shown that the addition of AFA considerably increases the proof theoretic strength.
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  9.  27
    Set Theory and Its Logic.J. C. Shepherdson & Willard Van Orman Quine - 1965 - Philosophical Quarterly 15 (61):371.
  10. 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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  11.  11
    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.  7
    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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  13.  7
    Set Theory and its Logic, Revised Edition. [REVIEW]P. K. H. - 1970 - Review of Metaphysics 23 (3):563-564.
    This revision of an important and lucid account of the various systems of axiomatic set theory preserves the basic format and essential ingredients of its highly regarded original. Quine's innovative exploitation of the virtual theory of classes in order to develop a considerable portion of set theory without ontological commitment to the existence of classes remains unchanged. So, too, does the list of topics treated--the theory of sets up to transfinite ordinal and cardinal numbers, the axiom (...)
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  14.  5
    Set Theory.Keith J. Devlin - 1981 - Journal of Symbolic Logic 46 (4):876-877.
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  15. Set Theory.T. Jech - 2005 - Bulletin of Symbolic Logic 11 (2):243-245.
     
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  16.  72
    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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  17. Modal Set Theory.Christopher Menzel - forthcoming - In Otávio Bueno & Scott 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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  18.  27
    Set Theory.Thomas Jech - 1981 - Journal of Symbolic Logic.
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  19. Constructive Set Theory.John Myhill - 1975 - Journal of Symbolic Logic 40 (3):347-382.
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  20.  11
    Set Theory. An Introduction to Independence Proofs.James E. Baumgartner & Kenneth Kunen - 1986 - Journal of Symbolic Logic 51 (2):462.
  21. Cantorian Set Theory and Limitation of Size.Michael Hallett - 1990 - Studia Logica 49 (2):283-284.
     
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  22. Set Theory.K. Kuratowski & A. Mostowski - 1971 - Philosophy of Science 38 (2):314-315.
     
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  23.  24
    Set Theory and the Continuum Hypothesis.Kenneth Kunen - 1966 - Journal of Symbolic Logic 35 (4):591-592.
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  24. 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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  25.  45
    A Formalization of Set Theory Without Variables.Alfred Tarski & Steven R. Givant - 1987 - American Mathematical Soc..
    Completed in 1983, this work culminates nearly half a century of the late Alfred Tarski's foundational studies in logic, mathematics, and the philosophy of science. Written in collaboration with Steven Givant, the book appeals to a very broad audience, and requires only a familiarity with first-order logic. It is of great interest to logicians and mathematicians interested in the foundations of mathematics, but also to philosophers interested in logic, semantics, algebraic logic, or the methodology of the deductive sciences, and to (...)
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  26. Cantorian Set Theory and Limitation of Size.Michael Hallett - 1986 - Mind 95 (380):523-528.
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  27. 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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  28. Maximality Principles in Set Theory.Luca Incurvati - 2017 - Philosophia Mathematica 25 (2):159-193.
    In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, (...)
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  29.  41
    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.  46
    Axiomatic Set Theory.Alfons Borgers - 1960 - Journal of Symbolic Logic 25 (3):277-278.
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  31.  34
    Set Theory. An Introduction to Large Cardinals.Azriel Levy - 1978 - Journal of Symbolic Logic 43 (2):384-384.
  32. Naive Set Theory.Paul R. Halmos & Patrick Suppes - 1961 - Synthese 13 (1):86-87.
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  33.  26
    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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  34.  9
    Descriptive Set Theory.Richard Mansfield - 1981 - Journal of Symbolic Logic 46 (4):874-876.
  35.  4
    Axiomatic Set Theory[REVIEW]Patrick Suppes - 1962 - Philosophical Review 71 (2):268-269.
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  36.  77
    Set Theory and Its Philosophy: A Critical Introduction.Stewart Shapiro - 2005 - Mind 114 (455):764-767.
  37. 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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  38.  34
    Naive Set Theory and Nontransitive Logic.David Ripley - 2015 - Review of Symbolic Logic 8 (3):553-571.
  39.  75
    Rudimentary and Arithmetical Constructive Set Theory.Peter Aczel - 2013 - Annals of Pure and Applied Logic 164 (4):396-415.
    The aim of this paper is to formulate and study two weak axiom systems for the conceptual framework of constructive set theory . Arithmetical CST is just strong enough to represent the class of von Neumann natural numbers and its arithmetic so as to interpret Heyting Arithmetic. Rudimentary CST is a very weak subsystem that is just strong enough to represent a constructive version of Jensenʼs rudimentary set theoretic functions and their theory. The paper is a contribution to (...)
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  40. 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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  41. 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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  42. Higher Set Theory.Harvey Friedman - manuscript
    Russell’s way out of his paradox via the impre-dicative theory of types has roughly the same logical power as Zermelo set theory - which supplanted it as a far more flexible and workable axiomatic foundation for mathematics. We discuss some new formalisms that are conceptually close to Russell, yet simpler, and have the same logical power as higher set theory - as represented by the far more powerful Zermelo-Frankel set theory and beyond. END.
     
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  43.  26
    Set Theory and Its Logic.Joseph S. Ullian & Willard Van Orman Quine - 1966 - Philosophical Review 75 (3):383.
  44.  83
    Classes and Truths in Set Theory.Kentaro Fujimoto - 2012 - Annals of Pure and Applied Logic 163 (11):1484-1523.
    This article studies three most basic systems of truth as well as their subsystems over set theory ZF possibly with AC or the axiom of global choice GC, and then correlates them with subsystems of Morse–Kelley class theory MK. The article aims at making an initial step towards the axiomatic study of truth in set theory in connection with class theory. Some new results on the side of class theory, such as conservativity, forcing and some (...)
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  45.  9
    Naive Set Theory.Axiomatic Set Theory.Elliott Mendelson - 1960 - Journal of Philosophy 57 (15):512-513.
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  46.  37
    Basic Set Theory[REVIEW]H. T. Hodes - 1981 - Philosophical Review 90 (2):298-300.
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  47.  50
    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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  48.  4
    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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  49.  46
    Aspects of General Topology in Constructive Set Theory.Peter Aczel - 2006 - Annals of Pure and Applied Logic 137 (1-3):3-29.
    Working in constructive set theory we formulate notions of constructive topological space and set-generated locale so as to get a good constructive general version of the classical Galois adjunction between topological spaces and locales. Our notion of constructive topological space allows for the space to have a class of points that need not be a set. Also our notion of locale allows the locale to have a class of elements that need not be a set. Class sized mathematical structures (...)
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  50.  34
    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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