Search results for 'mathematical definitions' (try it on Scholar)

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  1. Christopher Belanger (2013). On Two Mathematical Definitions of Observational Equivalence: Manifest Isomorphism and Reconsidered. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 44 (2):69-76.score: 150.0
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  2. Jamie Tappenden (2008). Mathematical Concepts and Definitions. In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oup Oxford. 256--275.score: 126.0
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  3. S. C. Kleene (1949). Review: K. R. Popper, On the Theory of Deduction, Part I. Derivation and its Generalizations; K. R. Popper, On the Theory of Deduction, Part II. The Definitions of Classical and Intuitionist Negation; K. R. Popper, The Trivialization of Mathematical Logic. [REVIEW] Journal of Symbolic Logic 14 (1):62-63.score: 120.0
  4. Jan Mycielski (2004). On the Tension Between Tarski's Nominalism and His Model Theory (Definitions for a Mathematical Model of Knowledge). Annals of Pure and Applied Logic 126 (1-3):215-224.score: 120.0
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  5. Marwan Rashed (1996). Ex Aequali Ratios in the Greek and Arabic Euclidean Traditions Gregg De Young Euclid Discusses the Ex Aequali Relationship Twice in the Elements. The First is in Book V (Based on Definitions 17 and 18, Propositions 22 and 23), During His Discussion of Arithmetical Relations Between Mathematical Magnitudes In. [REVIEW] Arabic Sciences and Philosophy 6.score: 120.0
  6. Robert McNaughton (1997). Robinson Raphael M.. Restricted Set-Theoretical Definitions in Arithmetic. Proceedings of the American Mathematical Society, Vol. 9 (1958), Pp. 238–242. Robinson Raphael M.. Restricted Set-Theoretical Definitions in Arithmetic. Summaries of Talks Presented at the Summer Institute for Symbolic Logic, Cornell University, 1957, 2nd Edn., Communications Research Division, Institute for Defense Analyses, Princeton, NJ, 1960, Pp. 139–140. [REVIEW] Journal of Symbolic Logic 31 (4):659-660.score: 120.0
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  7. Solomon Feferman, Is the Continuum Hypothesis a Definite Mathematical Problem?score: 60.0
    The purpose of this article is to explain why I believe that the Continuum Hypothesis (CH) is not a definite mathematical problem. My reason for that is that the concept of arbitrary set essential to its formulation is vague or underdetermined and there is no way to sharpen it without violating what it is supposed to be about. In addition, there is considerable circumstantial evidence to support the view that CH is not definite.
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  8. Sanford Shieh (2009). Teaching & Learning Guide For: Frege on Definitions. Philosophy Compass 4 (5):885-888.score: 60.0
    Three clusters of philosophically significant issues arise from Frege's discussions of definitions. First, Frege criticizes the definitions of mathematicians of his day, especially those of Weierstrass and Hilbert. Second, central to Frege's philosophical discussion and technical execution of logicism is the so-called Hume's Principle, considered in The Foundations of Arithmetic . Some varieties of neo-Fregean logicism are based on taking this principle as a contextual definition of the operator 'the number of …', and criticisms of such neo-Fregean programs (...)
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  9. Charlotte Werndl (2009). Justifying Definitions in Mathematics—Going Beyond Lakatos. Philosophia Mathematica 17 (3):313-340.score: 60.0
    This paper addresses the actual practice of justifying definitions in mathematics. First, I introduce the main account of this issue, namely Lakatos's proof-generated definitions. Based on a case study of definitions of randomness in ergodic theory, I identify three other common ways of justifying definitions: natural-world justification, condition justification, and redundancy justification. Also, I clarify the interrelationships between the different kinds of justification. Finally, I point out how Lakatos's ideas are limited: they fail to show how (...)
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  10. Michael Rathjen (1996). Monotone Inductive Definitions in Explicit Mathematics. Journal of Symbolic Logic 61 (1):125-146.score: 60.0
    The context for this paper is Feferman's theory of explicit mathematics, T 0 . We address a problem that was posed in [6]. Let MID be the principle stating that any monotone operation on classifications has a least fixed point. The main objective of this paper is to show that T 0 + MID, when based on classical logic, also proves the existence of non-monotone inductive definitions that arise from arbitrary extensional operations on classifications. From the latter we deduce (...)
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  11. Uri Nodelman & Edward N. Zalta (2014). Foundations for Mathematical Structuralism. Mind 123 (489):39-78.score: 58.0
    We investigate the form of mathematical structuralism that acknowledges the existence of structures and their distinctive structural elements. This form of structuralism has been subject to criticisms recently, and our view is that the problems raised are resolved by proper, mathematics-free theoretical foundations. Starting with an axiomatic theory of abstract objects, we identify a mathematical structure as an abstract object encoding the truths of a mathematical theory. From such foundations, we derive consequences that address the main questions (...)
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  12. Demetra Christopoulou (2014). Weyl on Fregean Implicit Definitions: Between Phenomenology and Symbolic Construction. Journal for General Philosophy of Science 45 (1):35-47.score: 58.0
    This paper aims to investigate certain aspects of Weyl’s account of implicit definitions. The paper takes under consideration Weyl’s approach to a certain kind of implicit definitions i.e. abstraction principles introduced by Frege.ion principles are bi-conditionals that transform certain equivalence relations into identity statements, defining thereby mathematical terms in an implicit way. The paper compares the analytic reading of implicit definitions offered by the Neo-Fregean program with Weyl’s account which has phenomenological leanings. The paper suggests that (...)
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  13. Paul Cortois (1996). The Structure of Mathematical Experience According to Jean Cavaillèst. Philosophia Mathematica 4 (1):18-41.score: 54.0
    In this expository article one of the contributions of Jean Cavailles to the philosophy of mathematics is presented: the analysis of ‘mathematical experience’. The place of Cavailles on the logico-philosophical scene of the 30s and 40s is sketched. I propose a partial interpretation of Cavailles's epistemological program of so-called ‘conceptual dialectics’: mathematical holism, duality principles, the notion of formal contents, and the specific temporal structure of conceptual dynamics. The structure of mathematical abstraction is analysed in terms of (...)
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  14. R. Eugene Collins (2005). The Mathematical Basis for Physical Laws. Foundations of Physics 35 (5):743-785.score: 54.0
    Laws of mechanics, quantum mechanics, electromagnetism, gravitation and relativity are derived as “related mathematical identities” based solely on the existence of a joint probability distribution for the position and velocity of a particle moving on a Riemannian manifold. This probability formalism is necessary because continuous variables are not precisely observable. These demonstrations explain why these laws must have the forms previously discovered through experiment and empirical deduction. Indeed, the very existence of electric, magnetic and gravitational fields is predicted by (...)
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  15. S. Friederich (2011). Motivating Wittgenstein's Perspective on Mathematical Sentences as Norms. Philosophia Mathematica 19 (1):1-19.score: 54.0
    The later Wittgenstein’s perspective on mathematical sentences as norms is motivated for sentences belonging to Hilbertian axiomatic systems where the axioms are treated as implicit definitions. It is shown that in this approach the axioms are employed as norms in that they function as standards of what counts as using the concepts involved. This normative dimension of their mode of use, it is argued, is inherited by the theorems derived from them. Having been motivated along these lines, Wittgenstein’s (...)
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  16. A. T. Balaban (2005). Reflections About Mathematical Chemistry. Foundations of Chemistry 7 (3):289-306.score: 54.0
    A personal account is presented for the present status of mathematical chemistry, with emphasis on non-numerical applications. These use mainly graph-theoretical concepts. Most computational chemical applications involve quantum chemistry and are therefore largely reducible to physics, while discrete mathematical applications often do not. A survey is provided for opinions and definitions of mathematical chemistry, and then for journals, books and book series, as well as symposia of mathematical chemistry.
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  17. Reinhard Kahle (2002). Mathematical Proof Theory in the Light of Ordinal Analysis. Synthese 133 (1/2):237 - 255.score: 54.0
    We give an overview of recent results in ordinal analysis. Therefore, we discuss the different frameworks used in mathematical proof-theory, namely "subsystem of analysis" including "reverse mathematics", "Kripke-Platek set theory", "explicit mathematics", "theories of inductive definitions", "constructive set theory", and "Martin-Löf's type theory".
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  18. Matthew Donald (1995). A Mathematical Characterization of the Physical Structure of Observers. Foundations of Physics 25 (4):529-571.score: 54.0
    It is proposed that the physical structure of an observer in quantum mechanics is constituted by a pattern of elementary localized switching events. A key preliminary step in giving mathematical expression to this proposal is the introduction of an equivalence relation on sequences of spacetime sets which relates a sequence to any other sequence to which it can be deformed without change of causal arrangement. This allows an individual observer to be associated with a finite structure. The identification of (...)
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  19. Thomas Forster (2007). Implementing Mathematical Objects in Set Theory. Logique Et Analyse 50 (197):79-86.score: 54.0
    In general little thought is given to the general question of how to implement mathematical objects in set theory. It is clear that—at various times in the past—people have gone to considerable lengths to devise implementations with nice properties. There is a litera- ture on the evolution of the Wiener-Kuratowski ordered pair, and a discussion by Quine of the merits of an ordered-pair implemen- tation that makes every set an ordered pair. The implementation of ordinals as Von Neumann ordinals (...)
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  20. Steven Kieffer, Jeremy Avigad & Harvey Friedman, A Language for Mathematical Knowledge Management.score: 54.0
    We argue that the language of Zermelo Fraenkel set theory with definitions and partial functions provides the most promising bedrock semantics for communicating and sharing mathematical knowledge. We then describe a syntactic sugaring of that language that provides a way of writing remarkably readable assertions without straying far from the set-theoretic semantics. We illustrate with some examples of formalized textbook definitions from elementary set theory and point-set topology. We also present statistics concerning the complexity of these (...), under various complexity measures. (shrink)
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  21. Arkady Plotnitsky (2006). A New Book of Numbers: On the Precise Definition of Quantum Variables and the Relationships Between Mathematics and Physics in Quantum Theory. [REVIEW] Foundations of Physics 36 (1):30-60.score: 52.0
    Following Asher Peres’s observation that, as in classical physics, in quantum theory, too, a given physical object considered “has a precise position and a precise momentum,” this article examines the question of the definition of quantum variables, and then the new type (as against classical physics) of relationships between mathematics and physics in quantum theory. The article argues that the possibility of the precise definition and determination of quantum variables depends on the particular nature of these relationships.
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  22. Barkley Rosser (1939). Definition by Induction in Quine's New Foundations for Mathematical Logic. Journal of Symbolic Logic 4 (2):80-81.score: 50.0
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  23. Paola Cantù (2010). Aristotle's Prohibition Rule on Kind-Crossing and the Definition of Mathematics as a Science of Quantities. Synthese 174 (2):225 - 235.score: 46.0
    The article evaluates the Domain Postulate of the Classical Model of Science and the related Aristotelian prohibition rule on kind-crossing as interpretative tools in the history of the development of mathematics into a general science of quantities. Special reference is made to Proclus’ commentary to Euclid’s first book of Elements , to the sixteenth century translations of Euclid’s work into Latin and to the works of Stevin, Wallis, Viète and Descartes. The prohibition rule on kind-crossing formulated by Aristotle in Posterior (...)
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  24. Volker Peckhaus (1999). 19th Century Logic Between Philosophy and Mathematics. Bulletin of Symbolic Logic 5 (4):433-450.score: 46.0
    The history of modern logic is usually written as the history of mathematical or, more general, symbolic logic. As such it was created by mathematicians. Not regarding its anticipations in Scholastic logic and in the rationalistic era, its continuous development began with George Boole's The Mathematical Analysis of Logic of 1847, and it became a mathematical subdiscipline in the early 20th century. This style of presentation cuts off one eminent line of development, the philosophical development of logic, (...)
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  25. Tyler Marghetis & Rafael Núñez (2013). The Motion Behind the Symbols: A Vital Role for Dynamism in the Conceptualization of Limits and Continuity in Expert Mathematics. Topics in Cognitive Science 5 (2):299-316.score: 46.0
    The canonical history of mathematics suggests that the late 19th-century “arithmetization” of calculus marked a shift away from spatial-dynamic intuitions, grounding concepts in static, rigorous definitions. Instead, we argue that mathematicians, both historically and currently, rely on dynamic conceptualizations of mathematical concepts like continuity, limits, and functions. In this article, we present two studies of the role of dynamic conceptual systems in expert proof. The first is an analysis of co-speech gesture produced by mathematics graduate students while proving (...)
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  26. John Mandel (1964/1984). The Statistical Analysis of Experimental Data. Dover.score: 42.0
    First half of book presents fundamental mathematical definitions, concepts and facts while remaining half deals with statistics primarily as an interpretive tool. Well-written text, numerous worked examples with step-by-step presentation. 116 tables.
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  27. S. Maracchia (1971). Plato And Russell On The Definition Of Mathematics. Scientia 106:216-223.score: 42.0
    Russell's appreciations of mathematics. It brings out that plato and russell used almost identical words in their recognition of the fact that mathematics, being a hypothetical and deductive science, is founded upon postulates which cannot be proved and therefore upon conventions; also that it provides a source of beauty and intellectual delight which reassuringly - though in apparent contrast to this view - conveys underlying truths. The article further displays russell's view of plato's love for mathematics and the progressive development, (...)
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  28. Erik Palmgren (1992). Type-Theoretic Interpretation of Iterated, Strictly Positive Inductive Definitions. Archive for Mathematical Logic 32 (2):75-99.score: 42.0
    We interpret intuitionistic theories of (iterated) strictly positive inductive definitions (s.p.-ID i′ s) into Martin-Löf's type theory. The main purpose being to obtain lower bounds of the proof-theoretic strength of type theories furnished with means for transfinite induction (W-type, Aczel's set of iterative sets or recursion on (type) universes). Thes.p.-ID i′ s are essentially the wellknownID i -theories, studied in ordinal analysis of fragments of second order arithmetic, but the set variable in the operator form is restricted to occur (...)
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  29. Benno van den Berg (2013). Non-Deterministic Inductive Definitions. Archive for Mathematical Logic 52 (1-2):113-135.score: 42.0
    We study a new proof principle in the context of constructive Zermelo-Fraenkel set theory based on what we will call “non-deterministic inductive definitions”. We give applications to formal topology as well as a predicative justification of this principle.
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  30. Peter Cholak, Alberto Marcone & Reed Solomon (2004). Reverse Mathematics and the Equivalence of Definitions for Well and Better Quasi-Orders. Journal of Symbolic Logic 69 (3):683-712.score: 40.0
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  31. Joan Rand Moschovakis (1970). Review: Luitzen Egbertus Jan Brouwer, Stefan Bauer-Mangelberg, Jean van Heijenoort, On the Significance of the Principle of Excluded Middle in Mathematics, Especially in Function Theory; Luitzen Egbertus Jan Brouwer, Stefan Bauer-Mengelberg, On the Domains of Definition of Functions; Luitzen Egbertus Jan Brouwer, Stefan Bauer-Mangelberg, Intuitionistic Reflections on Formalism. [REVIEW] Journal of Symbolic Logic 35 (2):332-333.score: 40.0
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  32. Paul Bernays (1940). Review: Barkley Rosser, Definition by Indication in Quine's New Foundations for Mathematical Logic. [REVIEW] Journal of Symbolic Logic 5 (1):33-33.score: 40.0
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  33. Samuel R. Buss & Alan S. Johnson (2010). The Quantifier Complexity of Polynomial‐Size Iterated Definitions in First‐Order Logic. Mathematical Logic Quarterly 56 (6):573-590.score: 40.0
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  34. H. B. Curry (1954). Remarks on the Definition and Nature of Mathematics. Dialectica 8 (3):228-233.score: 40.0
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  35. Ylannis N. Moschovakis (1996). Lacombe Daniel. Quelques Procédés de Définition En Topologie Récursive. Constructivity in Mathematics, Proceedings of the Colloquium Held at Amsterdam, 1957, Edited by Heyting A., Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam 1959, Pp. 129–158. [REVIEW] Journal of Symbolic Logic 31 (1):133-134.score: 40.0
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  36. Ralf Schindler (2004). NCA Da Costa and FA Doria. Consequences of an Exotic Definition for P= NP. Applied Mathematics and Computation, Vol. 145 (2003), Pp. 655–665. [REVIEW] Bulletin of Symbolic Logic 10 (1):118-119.score: 40.0
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  37. Alonzo Church (1957). Review: H. B. Curry, Remarks on the Definition and Nature of Mathematics. [REVIEW] Journal of Symbolic Logic 22 (1):85-86.score: 40.0
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  38. T. Glaß, M. Rathjen & A. Schlüter (1997). The Strength of Monotone Inductive Definitions in Explicit Mathematics. Annals of Pure and Applied Logic 85:1-46.score: 40.0
     
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  39. Alexander Khait (2005). The Definition of Mathematics: Philosophical and Pedagogical Aspects. Science and Education 14 (2):137-159.score: 40.0
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  40. J. C. C. McKinsey (1940). Review: H. B. Curry, Remarks on the Definition and Nature of Mathematics. [REVIEW] Journal of Symbolic Logic 5 (1):26-26.score: 40.0
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  41. Helmut Pfeiffer (1994). Feferman S. And Sieg W. Inductive Definitions and Subsystems of Analysis. Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies, by Buchholz Wilfried, Feferman Solomon, Pohlers Wolfram, and Sieg Wilfried. Lecture Notes in Mathematics, Vol. 897, Springer-Verlag, Berlin, Heidelberg, and New York, 1981, Pp. 16–77. Feferman Solomon and Sieg Wilfried. Proof Theoretic Equivalences Between Classical and Constructive Theories for Analysis. Iterated Inductive Definitions and ... [REVIEW] Journal of Symbolic Logic 59 (2):668-670.score: 40.0
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  42. U. Roth (1997). The Definition of Mathematics in Nicholas of Cusa and Leibniz. Studia Leibnitiana 29 (1):63-80.score: 40.0
     
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  43. James Robert Brown (1999). Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures. Routledge.score: 38.0
    Philosophy of Mathematics is clear and engaging, and student friendly The book discusses the great philosophers and the importance of mathematics to their thought. Among topics discussed in the book are the mathematical image, platonism, picture-proofs, applied mathematics, Hilbert and Godel, knots and notation definitions, picture-proofs and Wittgenstein, computation, proof and conjecture.
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  44. Kajsa Bråting (2012). Ambiguities of Fundamental Concepts in Mathematical Analysis During the Mid-Nineteenth Century. Foundations of Science 17 (4):301-320.score: 38.0
    In this paper we consider the major development of mathematical analysis during the mid-nineteenth century. On the basis of Jahnke’s (Hist Math 20(3):265–284, 1993 ) distinction between considering mathematics as an empirical science based on time and space and considering mathematics as a purely conceptual science we discuss the Swedish nineteenth century mathematician E.G. Björling’s general view of real- and complexvalued functions. We argue that Björling had a tendency to sometimes consider mathematical objects in a naturalistic way. One (...)
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  45. Andrew Aberdein (2013). Mathematical Wit and Mathematical Cognition. Topics in Cognitive Science 5 (2):231-250.score: 38.0
    The published works of scientists often conceal the cognitive processes that led to their results. Scholars of mathematical practice must therefore seek out less obvious sources. This article analyzes a widely circulated mathematical joke, comprising a list of spurious proof types. An account is proposed in terms of argumentation schemes: stereotypical patterns of reasoning, which may be accompanied by critical questions itemizing possible lines of defeat. It is argued that humor is associated with risky forms of inference, which (...)
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  46. Andrea Cantini (1996). Logical Frameworks for Truth and Abstraction: An Axiomatic Study. Elsevier Science B.V..score: 36.0
    This English translation of the author's original work has been thoroughly revised, expanded and updated. The book covers logical systems known as type-free or self-referential . These traditionally arise from any discussion on logical and semantical paradoxes. This particular volume, however, is not concerned with paradoxes but with the investigation of type-free sytems to show that: (i) there are rich theories of self-application, involving both operations and truth which can serve as foundations for property theory and formal semantics; (ii) these (...)
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  47. Victor Pambuccian (2002). On Definitions in an Infinitary Language. Mathematical Logic Quarterly 48 (4):522-524.score: 36.0
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  48. Richard A. Shore (2007). Direct and Local Definitions of the Turing Jump. Journal of Mathematical Logic 7 (02):229-262.score: 36.0
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  49. Alexander Abian (1968). On Definitions of Cuts and Completion of Partially Ordered Sets. Mathematical Logic Quarterly 14 (19):299-302.score: 36.0
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  50. W. Buchholz (forthcoming). Proof Theory of Iterated Inductive Definitions Revisited. Archive for Mathematical Logic.score: 36.0
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