Results for 'Set Theory'

929 found
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  1.  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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  2.  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.
  3.  95
    The Type Theoretic Interpretation of Constructive Set Theory.Peter Aczel, Angus Macintyre, Leszek Pacholski & Jeff Paris - 1984 - Journal of Symbolic Logic 49 (1):313-314.
  4.  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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  5.  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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  6. Labyrinth of Thought. A History of Set Theory and Its Role in Modern Mathematics.José Ferreirós - 2002 - Studia Logica 72 (3):437-440.
     
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  7.  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.
  8. 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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  9.  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.
  10. On self-membered sets in Quine's set theory NF.Maurice Boffa & André Pétry - 1993 - Logique Et Analyse 141:142.
     
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  11. Trading Ontology for Ideology. The Interplay of Logic, Set Theory and Semantics in Quine's Philosophy.Lieven Decock - 2004 - Tijdschrift Voor Filosofie 66 (2):370-371.
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  12. Filozofska teorija skupova-Michael Potter: Set theory and its philosophy: A critical introduction, Oxford University Press, Oxford, 2004.Miloš Adžić - 2010 - Theoria: Beograd 53 (2):127-132.
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  13. Comments on the Foundations of Set Theory.Paul J. Cohen - 1975 - Journal of Symbolic Logic 40 (3):459-460.
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  14. A nominalistic proof of the conservativeness of set theory.Hartry Field - 1992 - Journal of Philosophical Logic 21 (2):111 - 123.
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  15.  47
    Questions and answers–a category arising in linear logic, complexity theory, and set theory.Andreas Blass - 1995 - In Jean-Yves Girard, Yves Lafont & Laurent Regnier, Advances in linear logic. New York, NY, USA: Cambridge University Press. pp. 222--61.
  16.  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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  17.  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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  18.  43
    Some Impredicative Definitions in the Axiomatic Set-Theory.Andrzej Mostowski - 1951 - Journal of Symbolic Logic 16 (4):274-275.
  19. 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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  20. 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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  21.  44
    The Set-theoretic Multiverse : A Natural Context for Set Theory.Joel David Hamkins - 2011 - Annals of the Japan Association for Philosophy of Science 19:37-55.
  22. 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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  23.  36
    The Fraenkel-Mostowski Method for Independence Proofs in Set Theory.J. W. Addison, Leon Henkin, Alfred Tarski & Paul E. Howard - 1975 - Journal of Symbolic Logic 40 (4):631-631.
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  24.  18
    Some Problems and Results relevant to the Foundations of Set Theory.Alfred Tarski & W. Hanf - 1965 - Journal of Symbolic Logic 30 (1):95-96.
  25.  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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  26.  89
    Natural models of Ackermann's set theory.Rudolf Grewe - 1969 - Journal of Symbolic Logic 34 (3):481-488.
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  27.  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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  28.  89
    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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  29.  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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  30. 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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  31.  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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  32.  16
    On Models of Zermelo-Fraenkel Set Theory Satisfying the Axiom of Constructibility.Andrzej Mostowski - 1971 - Journal of Symbolic Logic 36 (3):542-542.
  33.  27
    Constructive Set Theory with Operations.Andrea Cantini & Laura Crosilla - 2007 - In Alessandro Andretta, Keith Kearnes & Domenico Zambella, Logic Colloquium 2004: Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic, Held in Torino, Italy, July 25-31, 2004. Cambridge: Cambridge University Press.
    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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  34.  18
    Zermelo (1930) is concerned with impredicative second-order set theory. He treats the general case of set theory with urelements, but it will be enough to consider only the case of pure set theory, ie without urelements. In this context, Zermelo's theory is the axiomatic second-order theory T2 in the language of pure set theory whose axioms are Extensionality, Regu. [REVIEW]Ww Tait - 1998 - In Matthias Schirn, The Philosophy of Mathematics Today: Papers From a Conference Held in Munich From June 28 to July 4,1993. Oxford, England: Clarendon Press. pp. 469.
  35.  77
    Sharon Berry.*A Logical Foundation for Potentialist Set Theory.Chris Scambler - 2023 - Philosophia Mathematica 31 (2):277-282.
    This book offers a foundation for mathematics grounded in a collection of axioms for logical possibility in a first-order language. The offered foundation is ar.
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  36.  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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  37. 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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  38. 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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  39. 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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  40.  18
    The Consistency of the Axiom of Choice and of the Generalized Continuum- Hypothesis with the Axioms of Set Theory.George W. Brown - 1941 - Journal of Symbolic Logic 6 (3):112-114.
  41.  17
    Simplified Independence Proofs. Boolean Valued Models of Set Theory.J. Barkley Rosser - 1974 - Journal of Symbolic Logic 39 (2):328-329.
  42.  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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  43.  31
    John Tucker. The formalisation of set theory. Mind, n.s. vol. 72 , pp. 500–518.Richard Montague - 1966 - Journal of Symbolic Logic 31 (4):676.
  44. (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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  45.  36
    The Notion of Rank in Set-Theory.Dana Scott - 1966 - Journal of Symbolic Logic 31 (4):662-663.
  46.  64
    A completeness theorem for Zermelo-Fraenkel set theory.William C. Powell - 1976 - Journal of Symbolic Logic 41 (2):323-327.
  47.  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.
  48. Defending the axioms: On the philosophical foundations of set theory * by Penelope Maddy.S. Vineberg - 2012 - Analysis 72 (3):635-637.
  49.  25
    MORRIS, Sean: Quine, New Foundations, and the Philosophy of Set Theory.Ádám Tamas Tuboly - 2019 - Filozofia 74 (6).
  50.  46
    Combinator realizability of a constructive Morse set theory.John Staples - 1974 - Journal of Symbolic Logic 39 (2):226-234.
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