Results for 'Metamathematics'

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  1.  60
    Introduction to Metamathematics.Stephen Cole Kleene - 1952 - North Holland.
  2.  20
    The Mathematics of Metamathematics.Helena Rasiowa - 1963 - Warszawa, Państwowe Wydawn. Naukowe.
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  3.  28
    Logic, Semantics, Metamathematics Papers From 1923 to 1938.Alfred Tarski & John Corcoran (eds.) - 1983 - Hackett.
    Published with the aid of a grant from the National Endowment for the Humanities. Contains the only complete English-language text of The Concept of Truth in Formalized Languages. Tarski made extensive corrections and revisions of the original translations for this edition, along with new historical remarks. It includes a new preface and a new analytical index for use by philosophers and linguists as well as by historians of mathematics and philosophy.
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  4.  25
    Logic, Semantics, Metamathematics; Papers From 1923 to 1938.Alfred Tarski & J. H. Woodger - 1958 - Journal of Philosophy 55 (8):351-352.
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  5.  3
    Mechanism, Mentalism, and Metamathematics: An Essay on Finitism.Judson C. Webb - 1980 - Journal of Philosophy 81 (8):456-464.
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  6. The Metamathematics of Algebraic Systems, Collected Papers: 1936-1967.A. I. Malʹt͡sev - 1971 - Amsterdam: North-Holland Pub. Co..
  7. Logic, Semantics, Metamathematics Papers From 1923 to 1938.Tarski Alfred & J. H. Woodger - 1956 - Clarendon Press.
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  8. The Metamathematics of Putnam's Model-Theoretic Arguments.Tim Button - 2011 - Erkenntnis 74 (3):321-349.
    Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to metamathematical challenges.
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  9.  4
    The Metamathematics of Scattered Linear Orderings.P. Clote - 1989 - Archive for Mathematical Logic 29 (1):9-20.
    Pursuing the proof-theoretic program of Friedman and Simpson, we begin the study of the metamathematics of countable linear orderings by proving two main results. Over the weak base system consisting of arithmetic comprehension, II 1 1 -CA0 is equivalent to Hausdorff's theorem concerning the canonical decomposition of countable linear orderings into a sum over a dense or singleton set of scattered linear orderings. Over the same base system, ATR0 is equivalent to a version of the Continuum Hypothesis for linear (...)
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  10.  11
    'Metamathematics' in Transition.Matthias Wille - 2011 - History and Philosophy of Logic 32 (4):333 - 358.
    In this paper, we trace the conceptual history of the term ?metamathematics? in the nineteenth century. It is well known that Hilbert introduced the term for his proof-theoretic enterprise in about 1922. But he was verifiably inspired by an earlier usage of the phrase in the 1870s. After outlining Hilbert's understanding of the term, we will explore the lines of inducement and elucidate the different meanings of ?metamathematics? in the final decades of the nineteenth century. Finally, we will (...)
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  11.  17
    Boolos and the Metamathematics of Quine's Definitions of Logical Truth and Consequence.Günther Eder - 2016 - History and Philosophy of Logic 37 (2):170-193.
    The paper is concerned with Quine's substitutional account of logical truth. The critique of Quine's definition tends to focus on miscellaneous odds and ends, such as problems with identity. However, in an appendix to his influential article On Second Order Logic, George Boolos offered an ingenious argument that seems to diminish Quine's account of logical truth on a deeper level. In the article he shows that Quine's substitutional account of logical truth cannot be generalized properly to the general concept of (...)
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  12.  66
    Metamathematics and the Philosophy of Mind.Judson Webb - 1968 - Philosophy of Science 35 (June):156-78.
    The metamathematical theorems of Gödel and Church are frequently applied to the philosophy of mind, typically as rational evidence against mechanism. Using methods of Post and Smullyan, these results are presented as purely mathematical theorems and various such applications are discussed critically. In particular, J. Lucas's use of Gödel's theorem to distinguish between conscious and unconscious beings is refuted, while more generally, attempts to extract philosophy from metamathematics are shown to involve only dramatizations of the constructivity problem in foundations. (...)
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  13.  25
    On the Epistemological Justification of Hilbert's Metamathematics.Javier Legris - 2005 - Philosophia Scientiae 9 (2):225-238.
    The aim of this paper is to examine the idea of metamathematical deduction in Hilbert’s program showing its dependence of epistemological notions, specially the notion of intuitive knowledge. It will be argued that two levels of foundations of deduction can be found in the last stages of Hilbert’s Program. The first level is related to the reduction – in a particular sense – of mathematics to formal systems, which are ‘metamathematically’ justified in terms of symbolic manipulation. The second level of (...)
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  14.  64
    “Clarifying the Nature of the Infinite”: The Development of Metamathematics and Proof Theory.Jeremy Avigad - manuscript
    We discuss the development of metamathematics in the Hilbert school, and Hilbert’s proof-theoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.
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  15.  15
    Metamathematics and Philosophy.Jan Wolenski - 1983 - Bulletin of the Section of Logic 12 (4):221-225.
    The relevance of metamathematical researches for philosophy of math- ematics is an indubitable matter. In the paper I shall speak about impli- cations of metamathematics for general philosophy, especially for classical epistemological problems. Let us start with a historical observation con- cerning Hilbert's programme, the rst research programme in metamathe- matics as a separate study of formal systems. This programme was strongly in uence by epistemological considerations. In fact, Hilbert wanted to se- cure all classical mathematics against inconsistencies and (...)
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  16. Metamathematics, Machines, and Gödel's Proof.N. Shankar - 1994 - Cambridge University Press.
    The automatic verification of large parts of mathematics has been an aim of many mathematicians from Leibniz to Hilbert. While Gödel's first incompleteness theorem showed that no computer program could automatically prove certain true theorems in mathematics, the advent of electronic computers and sophisticated software means in practice there are many quite effective systems for automated reasoning that can be used for checking mathematical proofs. This book describes the use of a computer program to check the proofs of several celebrated (...)
     
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  17.  28
    The Metamathematics-Popperian Epistemology Connection and its Relation to the Logic of Turing's Programme.Jean-Roch Beausoleil - 1989 - British Journal for the Philosophy of Science 40 (3):307-322.
    Turing's programme, the idea that intelligence can be modelled computationally, is set in the context of a parallel between certain elements from metamathematics and Popper's schema for the evolution of knowledge. The parallel is developed at both the formal level, where it hinges on the recursive structuring of Popper's schema, and at the contentual level, where a few key issues common to both epistemology and metamathematics are briefly discussed. In light of this connection Popper's principle of transference, akin (...)
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  18.  26
    Nonmonotonicity in (the Metamathematics of) Arithmetic.Karl-Georg Niebergall - 1999 - Erkenntnis 50 (2-3):309-332.
    This paper is an attempt to bring together two separated areas of research: classical mathematics and metamathematics on the one side, non-monotonic reasoning on the other. This is done by simulating nonmonotonic logic through antitonic theory extensions. In the first half, the specific extension procedure proposed here is motivated informally, partly in comparison with some well-known non-monotonic formalisms. Operators V and, more generally, U are obtained which have some plausibility when viewed as giving nonmonotonic theory extensions. In the second (...)
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  19.  5
    Foundations for the Formalization of Metamathematics and Axiomatizations of Consequence Theories.Urszula Wybraniec-Skardowska - 2004 - Annals of Pure and Applied Logic 127 (1-3):243-266.
    This paper deals with Tarski's first axiomatic presentations of the syntax of deductive system. Andrzej Grzegorczyk's significant results which laid the foundations for the formalization of metalogic, are touched upon briefly. The results relate to Tarski's theory of concatenation, also called the theory of strings, and to Tarski's ideas on the formalization of metamathematics. There is a short mention of author's research in the field. The main part of the paper surveys research on the theory of deductive systems initiated (...)
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  20.  31
    Recursion Theory for Metamathematics.Raymond M. Smullyan - 1993 - Oxford University Press.
    This work is a sequel to the author's Godel's Incompleteness Theorems, though it can be read independently by anyone familiar with Godel's incompleteness theorem for Peano arithmetic. The book deals mainly with those aspects of recursion theory that have applications to the metamathematics of incompleteness, undecidability, and related topics. It is both an introduction to the theory and a presentation of new results in the field.
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  21. Logic, Semantics, Metamathematics.Alfred Tarski - 1956 - Oxford, Clarendon Press.
    I ON THE PRIMITIVE TERM OF LOGISTICf IN this article I propose to establish a theorem belonging to logistic concerning some connexions, not widely known, ...
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  22.  17
    Introduction to Model Theory and to the Metamathematics of Algebra.Abraham Robinson - 1963 - North-Holland.
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  23.  8
    Mechanism, Mentalism and Metamathematics.Judson Webb - 1980 - Kluwer Academic Publishers.
  24. Metamathematics of Comparability.Harvey Friedman - manuscript
    A number of comparability theorems have been investigated from the viewpoint of reverse mathematics. Among these are various comparability theorems between countable well orderings ([2],[8]), and between closed sets in metric spaces ([3],[5]). Here we investigate the reverse mathematics of a comparability theorem for countable metric spaces, countable linear orderings, and sets of rationals. The previous work on closed sets used a strengthened notion of continuous embedding. The usual weaker notion of continuous embedding is used here. As a byproduct, we (...)
     
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  25.  20
    Arithmetization of Metamathematics in a General Setting.S. Feferman - 1966 - Journal of Symbolic Logic 31 (2):269-270.
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  26. Metamathematics of Ulm Theory.Harvey Friedman - manuscript
    The classical Ulm theory provides a complete set of invariants for countable abelian p-groups, and hence also for countable torsion abelian groups. These invariants involve countable ordinals. One can read off many simple structural properties of such groups directly from the Ulm theory. We carry out a reverse mathematics analysis of several such properties. In many cases, we reverse to ATR0, thereby demonstrating a kind of necessary use of Ulm theory.
     
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  27.  42
    Dialectical Logic, Semantics and Metamathematics.Richard Routley - 1979 - Erkenntnis 14 (3):301 - 331.
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  28.  41
    The Metamathematics of Ergodic Theory.Jeremy Avigad - 2009 - Annals of Pure and Applied Logic 157 (2-3):64-76.
    The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theory provides rich opportunities for such analysis. Although the field has its origins in seventeenth century dynamics and nineteenth century statistical mechanics, it employs infinitary, nonconstructive, and structural methods that are characteristically modern. At the same time, computational concerns (...)
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  29.  12
    Incompleteness in Intuitionistic Metamathematics.David Charles McCarty - 1991 - Notre Dame Journal of Formal Logic 32 (3):323-358.
  30.  1
    Fragments of Kripke–Platek Set Theory and the Metamathematics of $$\Alpha $$ Α -Recursion Theory.Sy-David Friedman, Wei Li & Tin Lok Wong - 2016 - Archive for Mathematical Logic 55 (7-8):899-924.
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  31.  22
    From Metamathematics to Cognitive Science.Henrique de Morais Ribeiro - 1999 - Trans/Form/Ação 21 (1):181-193.
    In this article, it is suggested a possible profile for the historical and philosophical migration of several issues from the metamathematical domain to the domain of functionalist neuro-computational Cognitive Science. The description of such a transition is accomplished by an analysis of the ideas of Post, Church, Gödel, and, in particular, Turing on the possibility of formalization of creative thinking in Mathematics.Neste artigo, propõe-se uma configuração possível para a transição histórico-filosófica de temas investigados no domínio da metamatemática para o domínio (...)
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  32. Hilbert and Bernays on Metamathematics.P. Mancosu - 1998 - In ¸ Itemancosu1998. Oxford University Press. pp. 149--188.
     
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  33.  42
    Alfred Tarski: Semantic Shift, Heuristic Shift in Metamathematics.Hourya Sinaceur - 2001 - Synthese 126 (1-2):49 - 65.
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  34.  70
    Arithmetization of Metamathematics in a General Setting.Solomon Feferman with with R. L. Vaught - manuscript
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  35.  15
    The Metamathematics of Infinitary Set Theoretical Systems.Klaus Gloede - 1977 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 23 (1-6):19-44.
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  36.  18
    The Mathematics of Metamathematics[REVIEW]J. M. P. - 1965 - Review of Metaphysics 19 (1):157-157.
  37.  3
    On the Metamathematics of Algebra.Abraham Robinson - 1952 - Journal of Symbolic Logic 17 (3):205-207.
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  38.  4
    The Metamathematics of Random Graphs.John T. Baldwin - 2006 - Annals of Pure and Applied Logic 143 (1):20-28.
    We explain and summarize the use of logic to provide a uniform perspective for studying limit laws on finite probability spaces. This work connects developments in stability theory, finite model theory, abstract model theory, and probability. We conclude by linking this context with work on the Urysohn space.
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  39.  12
    Book Review. Mechanism, Mentalism and Metamathematics. J Webb. [REVIEW]Harold T. Hodes - 1984 - Journal of Philosophy 81 (8):456-64.
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  40.  3
    Mechanism, Mentalism, and Metamathematics. An Essay on Finitism.Stewart Shapiro & Judson Chambers Webb - 1980 - Journal of Symbolic Logic 51 (2):472.
  41.  12
    Introduction to Model Theory and to the Metamathematics of Algebra. [REVIEW]J. M. P. - 1965 - Review of Metaphysics 19 (1):157-158.
  42. Review: Abraham Robinson, On the Metamathematics of Algebra. [REVIEW]Leon Henkin - 1952 - Journal of Symbolic Logic 17 (3):205-207.
  43. Introduction to Model Theory and to the Metamathematics of Algebra.José Sanmartín Esplugues - 1971 - Teorema: International Journal of Philosophy 1 (4):134-136.
     
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  44.  11
    Logic, Semantics, Metamathematics; Papers From 1923 to 1938. [REVIEW]R. M. Martin - 1958 - Journal of Philosophy 55 (8):351-352.
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  45.  16
    Alfred Tarski's Work on General Metamathematics.W. J. Blok & Don Pigozzi - 1988 - Journal of Symbolic Logic 53 (1):36-50.
  46.  11
    Metamathematics of Fuzzy Logic.Francis Jeffry Pelletier - 2000 - Bulletin of Symbolic Logic 6 (3):342-346.
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  47.  6
    Kleene Stephen Cole. Introduction to Metamathematics. North-Holland Publishing Co., Amsterdam, and P. Noordhoff, Groningen, 1952; D. Van Nostrand Company, New York and Toronto 1952; X + 550 Pp. [REVIEW]H. Rasiowa - 1954 - Journal of Symbolic Logic 19 (3):215-216.
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  48.  10
    Mechanism, Mentalism, and Metamathematics: An Essay on Finitism by Judson C. Webb. [REVIEW]Harold T. Hodes - 1984 - Journal of Philosophy 81 (8):456-464.
  49.  23
    Four Studies in Metamathematics.Stanisław J. Surma - 1968 - Studia Logica 23 (1):109-114.
  50.  11
    Review: Judson Chambers Webb, Mechanism, Mentalism, and Metamathematics. An Essay on Finitism. [REVIEW]Stewart Shapiro - 1986 - Journal of Symbolic Logic 51 (2):472-476.
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