Results for 'Hyperhypersimple set'

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  1.  23
    Hyperhypersimple sets and Q1 -reducibility.Irakli Chitaia - 2016 - Mathematical Logic Quarterly 62 (6):590-595.
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  2. On the orbits of hyperhypersimple sets.Wolfgang Maass - 1984 - Journal of Symbolic Logic 49 (1):51-62.
    This paper contributes to the question of under which conditions recursively enumerable sets with isomorphic lattices of recursively enumerable supersets are automorphic in the lattice of all recursively enumerable sets. We show that hyperhypersimple sets (i.e. sets where the recursively enumerable supersets form a Boolean algebra) are automorphic if there is a Σ 0 3 -definable isomorphism between their lattices of supersets. Lerman, Shore and Soare have shown that this is not true if one replaces Σ 0 3 by (...)
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  3.  22
    Hyperhypersimple sets and Δ2 systems.C. T. Chong - 1989 - Annals of Pure and Applied Logic 44 (1-2):25-38.
  4. Orbits of hyperhypersimple sets and the lattice of ∑03 sets.E. Herrmann - 1983 - Journal of Symbolic Logic 48 (3):693 - 699.
    It will be shown that in the lattice of recursively enumerable sets all lattices $\underline{L}(X)$ are elementarily definable with parameters, where X is Σ 0 3 and $\underline{L}^3(X)$ consists of all Σ 0 3 sets containing X.
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  5.  41
    (1 other version)A theorem on hyperhypersimple sets.Donald A. Martin - 1963 - Journal of Symbolic Logic 28 (4):273-278.
  6.  27
    Simplicity of Recursively Enumerable Sets.Two Theorems on hyperhypersimple Sets.On the Lattice of Recursively Enumerable Sets.The Elementary Theory of Recursively Enumerable Sets.Robert W. Robinson & A. H. Lachlan - 1970 - Journal of Symbolic Logic 35 (1):153-155.
  7.  34
    Robert W. Robinson. Simplicity of recursively enumerable sets.The journal of symbolic logic, vol. 32 , pp. 162–172. - Robert W. Robinson. Two theorems on hyperhypersimple sets. Transactions of the American Mathematical Society, vol. 128 , pp. 531–538. - A. H. Lachlan. On the lattice of recursively enumerable sets.Transactions of the American Mathematical Society, vol. 130 , pp. 1–37. - A. H. Lachlan. The elementary theory of recursively enumerable sets. Duke mathematical journal, vol. 35 , pp. 123–146. [REVIEW]James C. Owings - 1970 - Journal of Symbolic Logic 35 (1):153-155.
  8.  44
    Hyperhypersimple α-r.e. sets.C. T. Chong & M. Lerman - 1976 - Annals of Mathematical Logic 9 (1-2):1-48.
  9.  46
    Simple and hyperhypersimple vector spaces.Allen Retzlaff - 1978 - Journal of Symbolic Logic 43 (2):260-269.
    Let $V_\propto$ be a fixed, fully effective, infinite dimensional vector space. Let $\mathscr{L}(V_\propto)$ be the lattice consisting of the recursively enumerable (r.e.) subspaces of $V_\propto$ , under the operations of intersection and weak sum (see § 1 for precise definitions). In this article we examine the algebraic properties of $\mathscr{L}(V_\propto)$ . Early research on recursively enumerable algebraic structures was done by Rabin [14], Frolich and Shepherdson [5], Dekker [3], Hamilton [7], and Guhl [6]. Our results are based upon the more (...)
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  10.  27
    Q1-degrees of c.e. sets.R. Sh Omanadze & Irakli O. Chitaia - 2012 - Archive for Mathematical Logic 51 (5-6):503-515.
    We show that the Q-degree of a hyperhypersimple set includes an infinite collection of Q1-degrees linearly ordered under \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}Q1{\leq_{Q_1}}\end{document} with order type of the integers and consisting entirely of hyperhypersimple sets. Also, we prove that the c.e. Q1-degrees are not an upper semilattice. The main result of this paper is that the Q1-degree of a hemimaximal set contains only one c.e. 1-degree. Analogous results are valid for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} (...)
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  11.  48
    On the Degrees of Diagonal Sets and the Failure of the Analogue of a Theorem of Martin.Keng Meng Ng - 2009 - Notre Dame Journal of Formal Logic 50 (4):469-493.
    Semi-hyperhypersimple c.e. sets, also known as diagonals, were introduced by Kummer. He showed that by considering an analogue of hyperhypersimplicity, one could characterize the sets which are the Halting problem relative to arbitrary computable numberings. One could also consider half of splittings of maximal or hyperhypersimple sets and get another variant of maximality and hyperhypersimplicity, which are closely related to the study of automorphisms of the c.e. sets. We investigate the Turing degrees of these classes of c.e. sets. (...)
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  12.  50
    Some New Lattice Constructions in High R. E. Degrees.Heinrich Rolletschek - 1995 - Mathematical Logic Quarterly 41 (3):395-430.
    A well-known theorem by Martin asserts that the degrees of maximal sets are precisely the high recursively enumerable degrees, and the same is true with ‘maximal’ replaced by ‘dense simple’, ‘r-maximal’, ‘strongly hypersimple’ or ‘finitely strongly hypersimple’. Many other constructions can also be carried out in any given high r. e. degree, for instance r-maximal or hyperhypersimple sets without maximal supersets . In this paper questions of this type are considered systematically. Ultimately it is shown that every conjunction of (...)
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  13.  17
    r‐Maximal sets and Q1,N‐reducibility.Roland Sh Omanadze & Irakli O. Chitaia - 2021 - Mathematical Logic Quarterly 67 (2):138-148.
    We show that if M is an r‐maximal set, A is a major subset of M, B is an arbitrary set and, then. We prove that the c.e. ‐degrees are not dense. We also show that there exist infinite collections of ‐degrees and such that the following hold: (i) for every i, j,, and,(ii) each consists entirely of r‐maximal sets, and(iii) each consists entirely of non‐r‐maximal hyperhypersimple sets.
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  14.  13
    On Some Complexity Characteristics of Immune Sets.Valeriy K. Bulitko - 1995 - Mathematical Logic Quarterly 41 (3):307-313.
    We suggest some new ways to effectivize the definitions of several classes of simple sets. On this basis, new completeness criterions for simple sets are obtained. In particular, we give descriptions of the class of complete maximal sets.
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  15.  46
    Hyperhypersimple supersets in admissible recursion theory.C. T. Chong - 1983 - Journal of Symbolic Logic 48 (1):185-192.
  16.  33
    (1 other version)G. Metakides and A. Nerode. Recursion theory and algebra. Algebra and logic, Papers from the 1974 Summer Research Institute of the Australian Mathematical Society, Monash University, Australia, edited by J. N. Crossley, Lecture notes in mathematics, vol. 450, Springer-Verlag, Berlin, Heidelberg, and New York, 1975, pp. 209–219. - Iraj Kalantari and Allen Retzlaff. Maximal vector spaces under automorphisms of the lattice of recursively enumerable vector spaces. The journal of symbolic logic, vol. 42 no. 4 , pp. 481–491. - Iraj Kalantari. Major subspaces of recursively enumerable vector spaces. The journal of symbolic logic, vol. 43 , pp. 293–303. - J. Remmel. A r-maximal vector space not contained in any maximal vector space. The journal of symbolic logic, vol. 43 , pp. 430–441. - Allen Retzlaff. Simple and hyperhypersimple vector spaces. The journal of symbolic logic, vol. 43 , pp. 260–269. - J. B. Remmel. Maximal and cohesive vector spaces. The journal of symbolic logic, vol. 42 no. 3. [REVIEW]Henry A. Kierstead - 1986 - Journal of Symbolic Logic 51 (1):229-232.
  17.  10
    Funk utforsket.Lars Mjøset - 2013 - Agora Journal for metafysisk spekulasjon 31 (1-2):155-186.
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  18.  14
    Kina gjennom to globaliseringsperioder.Lars Mjøset & Rune Skarstein - 2017 - Agora Journal for metafysisk spekulasjon 34 (2-3):85-134.
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  19.  19
    Nyliberalisme, økonomisk teori og kapitalismens mangfold.Lars Mjøset - 2011 - Agora Journal for metafysisk spekulasjon 29 (1):54-93.
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  20. Nordic social theory Between social philosophy and grounded theory.Lars Mjøset - 2006 - In Gerard Delanty, The handbook of contemporary European social theory. New York: Routledge. pp. 123.
  21.  14
    Arquitetura vitruviana e retórica antiga.Settings Gilson Charles dos Santos - 2019 - Archai: Revista de Estudos Sobre as Origens Do Pensamento Ocidental 28:e02804.
    O objetivo deste artigo é apresentar a analogia básica entre arquitetura e retórica antiga a partir dos tratados De Architectura, de Vitrúvio, e o De Oratore, de Cícero. A analogia se verifica na definição do artífice, dos gêneros e partes das técnicas e dos fins de cada uma delas. Para tanto, tomaram-se como referência as fontes do tratado vitruviano, que menciona a influência de Varrão na gramática, de Lucrécio na filosofia e de Cícero no método oratório. A analogia com Cícero (...)
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  22. Multi-volume works in progress (1).Hist Set - forthcoming - History of Science.
     
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  23. Semester examinations–april 2013.Sem Set - 2011 - Business Ethics 4:10PBA4102.
     
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  24. Social order and the natural world.Hist Set - forthcoming - History of Science.
     
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  25. The Darwin Industry—A Critical Evalution.Hist Set - 1974 - History of Science 12:43.
     
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  26. Yogadarśana meṃ Īśvara praṇidhāna kī vyākhyā: Pātañjala-Yogadarśana.Anupamā Seṭha - 1994 - Dillī: Nāga Prakāśaka. Edited by Patañjali.
    Study, with text of the Yogasūtra of Patañjali, text on Yoga philosophy.
     
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  27. Herman Cappelen and Ernest Lepore.I. Stage Setting & Semantic Minimalism - 2004 - In R. Stanton, M. Ezcurdia & C. Viger, New Essays in Philosophy of Language and Mind, Canadian Journal of Philosophy, Supplementary Volume 30. University of Calgary Press. pp. 3.
     
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  28.  48
    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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  29. The modal logic of set-theoretic potentialism and the potentialist maximality principles.Joel David Hamkins & Øystein Linnebo - 2022 - Review of Symbolic Logic 15 (1):1-35.
    We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe [14], including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism, Grothendieck–Zermelo potentialism, transitive-set potentialism, forcing potentialism, countable-transitive-model potentialism, countable-model potentialism, and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, (...)
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  30. (1 other version)Cantorian Set Theory and Limitation of Size.Michael Hallett - 1986 - Mind 95 (380):523-528.
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  31.  55
    Defending the Axioms: On the Philosophical Foundations of Set Theory.Penelope Maddy - 2011 - Oxford, England: Oxford University Press.
    Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. For nearly a century, the axioms of set theory have played this role, so the question of how these axioms are properly judged takes on a central importance. Approaching the question from a broadly naturalistic or second-philosophical point of view, Defending the Axioms isolates the appropriate methods for such evaluations and investigates the ontological and epistemological backdrop that makes them appropriate. In the end, a new account of (...)
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  32.  74
    A Logical Foundation for Potentialist Set Theory.Sharon Berry - 2022 - Cambridge University Press.
    In many ways set theory lies at the heart of modern mathematics, and it does powerful work both philosophical and mathematical – as a foundation for the subject. However, certain philosophical problems raise serious doubts about our acceptance of the axioms of set theory. In a detailed and original reassessment of these axioms, Sharon Berry uses a potentialist approach to develop a unified determinate conception of set-theoretic truth that vindicates many of our intuitive expectations regarding set theory. Berry further defends (...)
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  33. Transfinite numbers in paraconsistent set theory.Zach Weber - 2010 - Review of Symbolic Logic 3 (1):71-92.
    This paper begins an axiomatic development of naive set theoryin a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead (...)
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  34.  50
    Which set existence axioms are needed to prove the separable Hahn-Banach theorem?Douglas K. Brown & Stephen G. Simpson - 1986 - Annals of Pure and Applied Logic 31:123-144.
  35.  16
    Lectures in set theory.Thomas J. Jech - 1971 - New York,: Springer Verlag.
  36. Transfinite Cardinals in Paraconsistent Set Theory.Zach Weber - 2012 - Review of Symbolic Logic 5 (2):269-293.
    This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem.
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  37. 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 forms of the reflection principle, (...)
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  38.  81
    Natural deduction based set theories: a new resolution of the old paradoxes.Paul C. Gilmore - 1986 - Journal of Symbolic Logic 51 (2):393-411.
    The comprehension principle of set theory asserts that a set can be formed from the objects satisfying any given property. The principle leads to immediate contradictions if it is formalized as an axiom scheme within classical first order logic. A resolution of the set paradoxes results if the principle is formalized instead as two rules of deduction in a natural deduction presentation of logic. This presentation of the comprehension principle for sets as semantic rules, instead of as a comprehension axiom (...)
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  39.  23
    Quine, New Foundations, and the Philosophy of Set Theory.Sean Morris - 2018 - New York: Cambridge University Press.
    Quine's set theory, New Foundations, has often been treated as an anomaly in the history and philosophy of set theory. In this book, Sean Morris shows that it is in fact well-motivated, emerging in a natural way from the early development of set theory. Morris introduces and explores the notion of set theory as explication: the view that there is no single correct axiomatization of set theory, but rather that the various axiomatizations all serve to explicate the notion of set (...)
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  40. (1 other version)Twist-Valued Models for Three-valued Paraconsistent Set Theory.Walter Carnielli & Marcelo E. Coniglio - 2021 - Logic and Logical Philosophy 30 (2):187-226.
    Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the axioms of (...)
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  41.  63
    Cantorian Set Theory and Limitation of Size. Michael Hallett.Robert Bunn - 1988 - Philosophy of Science 55 (3):461-478.
    The usual objections to infinite numbers, and classes, and series, and the notion that the infinite as such is self-contradictory, may... be dismissed as groundless. There remains, however, a very grave difficulty, connected with the contradiction [of the class of all classes not members of themselves]. This difficulty does not concern the infinite as such, but only certain very large infinite classes.
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  42. (1 other version)Iterative Conceptions of Set.Neil Barton - 2024 - Cambridge University Press.
    Many philosophers are aware of the paradoxes of set theory (e.g. Russell's paradox). For many people, these were solved by the iterative conception of set which holds that sets are formed in stages by collecting sets available at previous stages. This Element will examine possibilities for articulating this solution. In particular, the author argues that there are different kinds of iterative conception, and it's open which of them (if any) is the best. Along the way, the author hopes to make (...)
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  43.  90
    On Number-Set Identity: A Study.Sean C. Ebels-Duggan - 2022 - Philosophia Mathematica 30 (2):223-244.
    Benacerraf’s 1965 multiple-reductions argument depends on what I call ‘deferential logicism’: his necessary condition for number-set identity is most plausible against a background Quineanism that allows autonomy of the natural number concept. Steinhart’s ‘folkist’ sufficient condition on number-set identity, by contrast, puts that autonomy at the center — but fails for not taking the folk perspective seriously enough. Learning from both sides, we explore new conditions on number-set identity, elaborating a suggestion from Wright.
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  44. (1 other version)Second order logic or set theory?Jouko Väänänen - 2012 - Bulletin of Symbolic Logic 18 (1):91-121.
    We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each (...)
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  45.  3
    Crypto-preorders, topological relations, information and logic.Piero Pagliani International Rough Set Society, Rome & Italy - 2024 - Journal of Applied Non-Classical Logics 34 (2):330-367.
    As is well known, any preorder R on a set U induces an Alexandrov topology on U. In some interesting cases related to data mining an Alexandrov topology can be transformed into different types of logico-algebraic models. In some cases, (pre)topological operators provided by Pointless Topology may define a topological space on U even if R is not a preorder. If this is the case, then we call R a crypto-preorder. The paper studies the conditions under which a relation R (...)
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  46. A system of axiomatic set theory—Part I.Paul Bernays - 1937 - Journal of Symbolic Logic 2 (1):65-77.
    Introduction. The system of axioms for set theory to be exhibited in this paper is a modification of the axiom system due to von Neumann. In particular it adopts the principal idea of von Neumann, that the elimination of the undefined notion of a property (“definite Eigenschaft”), which occurs in the original axiom system of Zermelo, can be accomplished in such a way as to make the resulting axiom system elementary, in the sense of being formalizable in the logical calculus (...)
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  47. Worlds and Propositions Set Free.Otávio Bueno, Christopher Menzel & Edward N. Zalta - 2014 - Erkenntnis 79 (4):797–820.
    The authors provide an object-theoretic analysis of two paradoxes in the theory of possible worlds and propositions stemming from Russell and Kaplan. After laying out the paradoxes, the authors provide a brief overview of object theory and point out how syntactic restrictions that prevent object-theoretic versions of the classical paradoxes are justified philosophically. The authors then trace the origins of the Russell paradox to a problematic application of set theory in the definition of worlds. Next the authors show that an (...)
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  48.  87
    Fuzzy Empiricism and Fuzzy‐Set Causality: What Is All the Fuzz About?Jordi Cat - 2006 - Philosophy of Science 73 (1):26-41.
    This paper examines a novel notion of causality, namely, fuzzy-set-theoretic causality. Over the last decade, a number of conceptual models of causality, in the language of fuzzy-set theory, have appeared in the scientific literature and have been applied to empirical research. They have circulated widely from one scientific discipline to another, weaving a unifying thread through them. However, they have received no philosophical attention. In this paper, I will discuss the value and limitations of this type of model and will (...)
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  49. Cohen and set theory.Akihiro Kanamori - 2008 - Bulletin of Symbolic Logic 14 (3):351-378.
    We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing.
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  50.  21
    Models of ZF-set theory.Ulrich Felgner - 1971 - New York,: Springer Verlag.
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