Results for 'Set theory'

932 found
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  1.  42
    Generalizing realizability and Heyting models for constructive set theory.Albert Ziegler - 2012 - Annals of Pure and Applied Logic 163 (2):175-184.
  2.  29
    On the crispness of and arithmetic with a bisimulation in a constructive naive set theory.S. Yatabe - 2014 - Logic Journal of the IGPL 22 (3):482-493.
  3.  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.
  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. 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.  23
    Burgess on Plural Logic and Set Theory.O. Linnebo - 2007 - Philosophia Mathematica 15 (1):79-93.
  7. 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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  8.  80
    The ∀ n∃‐Completeness of Zermelo‐Fraenkel Set Theory.Daniel Gogol - 1978 - Mathematical Logic Quarterly 24 (19-24):289-290.
  9.  37
    An exact feature selection algorithm based on rough set theory.Mohammad Taghi Rezvan, Ali Zeinal Hamadani & Seyed Reza Hejazi - 2015 - Complexity 20 (5):50-62.
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  10.  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.
  11. 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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  12.  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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  13.  20
    Changing cofinalities and collapsing cardinals in models of set theory.Miloš S. Kurilić - 2003 - Annals of Pure and Applied Logic 120 (1-3):225-236.
    If a˜cardinal κ1, regular in the ground model M, is collapsed in the extension N to a˜cardinal κ0 and its new cofinality, ρ, is less than κ0, then, under some additional assumptions, each cardinal λ>κ1 less than cc/[κ1]<κ1) is collapsed to κ0 as well. If in addition N=M[f], where f : ρ→κ1 is an unbounded mapping, then N is a˜λ=κ0-minimal extension. This and similar results are applied to generalized forcing notions of Bukovský and Namba.
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  14.  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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  15.  33
    Bernays Paul. A system of axiomatic set theory — Part VII.J. R. Shoenfield - 1957 - Journal of Symbolic Logic 22 (4):367-368.
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  16.  45
    The Converse to a Metatheorem in Gödel Set Theory.Richard A. Platek - 1971 - Mathematical Logic Quarterly 17 (1):21-22.
  17.  81
    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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  18.  19
    Set Theory and a Model of the Mind in Psychology.Asger Törnquist & Jens Mammen - 2023 - Review of Symbolic Logic 16 (4):1233-1259.
    We investigate the mathematics of a model of the human mind which has been proposed by the psychologist Jens Mammen. Mathematical realizations of this model consists of what the first author (A.T.) has called Mammen spaces, where a Mammen space is a triple in the Baumgartner–Laver model.Finally, consequences for psychology are discussed.
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  19. 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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  20.  66
    A completeness theorem for Zermelo-Fraenkel set theory.William C. Powell - 1976 - Journal of Symbolic Logic 41 (2):323-327.
  21.  23
    (1 other version)Elementary Extensions of Models of the Alternative Set Theory.P. Pudlák & A. Sochor - 1985 - Mathematical Logic Quarterly 31 (19‐20):309-316.
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  22.  74
    Maddy, Penelope, Defending the Axioms: On the Philosophical Foundations of Set Theory, Oxford: Oxford University Press, 2011, pp. x + 150, £29/us$45.Jeffrey W. Roland - 2013 - Australasian Journal of Philosophy 91 (4):809-812.
  23.  52
    Set Theory and Its Logic.J. C. Shepherdson & Willard Van Orman Quine - 1965 - Philosophical Quarterly 15 (61):371.
  24.  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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  25.  19
    (1 other version)Contributions to the Theory of Semisets I. Relations of the theory of semisets to the Zermelo‐Fraenkel set theory.Petr Hájek - 1972 - Mathematical Logic Quarterly 18 (16‐18):241-248.
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  26. EVANS, DM, and HEWITT, PR, Counterexamples to a con-jecture on relative categoricity GOODMAN, ND, Topological models of epistemic set theory HEWITT, PR, see EVANS, DM.W. Hodges, Im Hodkinson & D. Macpherson - 1990 - Annals of Pure and Applied Logic 46:299.
  27.  9
    Some Applications of the Theory of Models to Set Theory.H. Jerome Keisler - 1967 - Journal of Symbolic Logic 32 (3):410-410.
  28.  50
    Superclasses in a Finite Extension of Zermelo Set Theory.Martin Kühnrich - 1978 - Mathematical Logic Quarterly 24 (31-36):539-552.
  29.  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.
  30. Idealist and Realist Elements in Cantor's Approach to Set Theory.I. Jane - 2010 - Philosophia Mathematica 18 (2):193-226.
    There is an apparent tension between the open-ended aspect of the ordinal sequence and the assumption that the set-theoretical universe is fully determinate. This tension is already present in Cantor, who stressed the incompletable character of the transfinite number sequence in Grundlagen and avowed the definiteness of the totality of sets and numbers in subsequent philosophical publications and in correspondence. The tension is particularly discernible in his late distinction between sets and inconsistent multiplicities. I discuss Cantor’s contrasting views, and I (...)
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  31. How set theory impinges on logic.Jesus Mosterin - unknown
    Standard (classical) logic is not independent of set theory. Which formulas are valid in logic depends on which sets we assume to exist in our set-theoretical universe. Second-order logic is just set theory in disguise. The typically logical notions of validity and consequence are not well defined in second-order logic, at least as long as there are open issues in set theory. Such contentious issues in set theory as the axiom of choice, the continuum hypothesis or (...)
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  32.  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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  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.  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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  35.  37
    Constructive notions of set: Part I. Sets in Martin–Löf type theory.Laura Crosilla - 2005 - Annali Del Dipartimento di Filosofia 11:347-387.
    This is the first of two articles dedicated to the notion of constructive set. In them we attempt a comparison between two different notions of set which occur in the context of the foundations for constructive mathematics. We also put them under perspective by stressing analogies and differences with the notion of set as codified in the classical theory Zermelo–Fraenkel. In the current article we illustrate in some detail the notion of set as expressed in Martin–L¨of type theory (...)
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  36.  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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  37.  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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  38.  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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  39. Causal sets and frame-valued set theory.John Bell - manuscript
    In spacetime physics any set C of events—a causal set—is taken to be partially ordered by the relation ≤ of possible causation: for p, q ∈ C, p ≤ q means that q is in p’s future light cone. In her groundbreaking paper The internal description of a causal set: What the universe looks like from the inside, Fotini Markopoulou proposes that the causal structure of spacetime itself be represented by “sets evolving over C” —that is, in essence, by the (...)
     
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  40.  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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  41. Rumfitt on the logic of set theory.Øystein Linnebo - 2019 - Inquiry: An Interdisciplinary Journal of Philosophy 62 (7):826-841.
    ABSTRACTAccording to a famous argument by Dummett, the concept of set is indefinitely extensible, and the logic appropriate for reasoning about the instances of any such concept is intuitionistic, not classical. But Dummett's argument is widely regarded as obscure. This note explains how the final chapter of Rumfitt's important new book advances our understanding of Dummett's argument, but it also points out some problems and unanswered questions. Finally, Rumfitt's reconstruction of Dummett's argument is contrasted with my own preferred alternative.
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  42. 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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  43.  83
    A Formalization of Set Theory Without Variables.István Németi - 1988 - 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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  44.  87
    Set Theory and Definite Descriptions.Karel Lambert - 2000 - Grazer Philosophische Studien 60 (1):1-11.
    This paper offers an explanation of the maj or traditions in the logical treatment of definite descriptions as reactions to paradoxical naive definite descriptiontheory. The explanation closely parallels that of various set theories as reactions to paradoxical naive set theory. Indeed, naive set theory is derivable from naive definite description theory given an appropriate definition of set abstracts in terms of definite descriptions.
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  45.  61
    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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  46.  31
    Proof Theory as an Analysis of Impredicativity( New Developments in Logic: Proof-Theoretic Ordinals and Set-Theoretic Ordinals).Ryota Akiyoshi - 2012 - Journal of the Japan Association for Philosophy of Science 39 (2):93-107.
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  47.  46
    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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  48. 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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  49.  20
    On the Growing Universe of Causal Set Theory—An Order-Type Approach.Tomasz Placek & Leszek Wroński - 2024 - Foundations of Physics 54 (3):1-30.
    We investigate a model of becoming—classical sequential growth (CSG)—that has been proposed within the framework of causal sets (causets), with the latter defined as order types of certain partial orderings. To investigate how causets grow, we introduce special sequences of causets, which we call “csg-paths”. We prove a number of results concerning relations between csg-paths and causets. These results paint a highly non-trivial picture of csg-paths. There are uncountably many csg-paths, all of them sharing the same beginning, after which they (...)
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  50.  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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