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  1. From Mathematics to Philosophy.Hao Wang - 1975 - British Journal for the Philosophy of Science 26 (2):170-174.
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  • Rosser sentences.D. Guaspari - 1979 - Annals of Mathematical Logic 16 (1):81.
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  • Hilbert's program relativized: Proof-theoretical and foundational reductions.Solomon Feferman - 1988 - Journal of Symbolic Logic 53 (2):364-384.
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  • Self-Reference and Modal Logic.George Boolos & C. Smorynski - 1988 - Journal of Symbolic Logic 53 (1):306.
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  • The Logic of Provability.George Boolos - 1993 - Cambridge and New York: Cambridge University Press.
    This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency. Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does (...)
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  • Aspects of Incompleteness.Per Lindström - 1999 - Studia Logica 63 (3):438-439.
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  • The Intrinsic Computational Difficulty of Functions.Alan Cobham - 1965 - In Yehoshua Bar-Hillel (ed.), Logic, methodology and philosophy of science. Amsterdam,: North-Holland Pub. Co.. pp. 24-30.
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  • An Unsolvable Problem of Elementary Number Theory.Alonzo Church - 1936 - Journal of Symbolic Logic 1 (2):73-74.
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  • Minds, Machines and Gödel.J. R. Lucas - 1961 - Etica E Politica 5 (1):1.
    In this article, Lucas maintains the falseness of Mechanism - the attempt to explain minds as machines - by means of Incompleteness Theorem of Gödel. Gödel’s theorem shows that in any system consistent and adequate for simple arithmetic there are formulae which cannot be proved in the system but that human minds can recognize as true; Lucas points out in his turn that Gödel’s theorem applies to machines because a machine is the concrete instantiation of a formal system: therefore, for (...)
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  • On Saying What You Really Want to Say: Wittgenstein, Gödel and the Trisection of the Angle.Juliet Floyd - 1995 - In Jaakko Hintikka (ed.), From Dedekind to Gödel: The Foundations of Mathematics in the Early Twentieth Century, Synthese Library Vol. 251 (Kluwer Academic Publishers. pp. 373-426.
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  • A Note on Boolos' Proof of the Incompleteness Theorem.Makoto Kikuchi - 1994 - Mathematical Logic Quarterly 40 (4):528-532.
    We give a proof of Gödel's first incompleteness theorem based on Berry's paradox, and from it we also derive the second incompleteness theorem model-theoretically.
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  • The Reception of Godel's Incompleteness Theorems.John W. Dawson - 1984 - PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1984:253 - 271.
    According to several commentators, Kurt Godel's incompleteness discoveries were assimilated promptly and almost without objection by his contemporaries - - a circumstance remarkable enough to call for explanation. Careful examination reveals, however, that there were doubters and critics, as well as defenders and rival claimants to priority. In particular, the reactions of Carnap, Bernays, Zermelo, Post, Finsler, and Russell, among others, are considered in detail. Documentary sources include unpublished correspondence from Godel's Nachlass.
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  • What is Mathematical Truth?Hilary Putnam - 1979 - In Philosophical Papers: Volume 1, Mathematics, Matter and Method. New York: Cambridge University Press. pp. 60--78.
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  • Wittgenstein's Philosophy of Mathematics.Michael Dummett - 1997 - Journal of Philosophy 94 (7):166--85.
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  • Peano's smart children: a provability logical study of systems with built-in consistency.Albert Visser - 1989 - Notre Dame Journal of Formal Logic 30 (2):161-196.
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  • Some Current Problems in Metamathematics 1.Alfred Tarski, Jan Tarski & Jan Woleński - 1995 - History and Philosophy of Logic 16 (2):159-168.
    In this article the author first described the developments which brought to focus the importance of consistency proofs for mathematics, and which led Hilbert to promote the science of metamathemat-ics. Further comments and remarks concern the (partly analogous) beginnings of the work on the decision problem, Gödel?s theorems and related matters, and general metamathematics. An appendix summarizes a text by the author on completeness and categoricity.
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  • Undefinability of truth. the problem of priority:tarski vs gödel.Roman Murawski - 1998 - History and Philosophy of Logic 19 (3):153-160.
    The paper is devoted to the discussion of some philosophical and historical problems connected with the theorem on the undefinability of the notion of truth. In particular the problem of the priority of proving this theorem will be considered. It is claimed that Tarski obtained this theorem independently though he made clear his indebtedness to Gödel’s methods. On the other hand, Gödel was aware of the formal undefinability of truth in 1931, but he did not publish this result. Reasons for (...)
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  • Beyond first-order logic: the historical interplay between mathematical logic and axiomatic set theory.Gregory H. Moore - 1980 - History and Philosophy of Logic 1 (1-2):95-137.
    What has been the historical relationship between set theory and logic? On the one hand, Zermelo and other mathematicians developed set theory as a Hilbert-style axiomatic system. On the other hand, set theory influenced logic by suggesting to Schröder, Löwenheim and others the use of infinitely long expressions. The questions of which logic was appropriate for set theory - first-order logic, second-order logic, or an infinitary logic - culminated in a vigorous exchange between Zermelo and Gödel around 1930.
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  • On Computable Numbers, with an Application to the Entscheidungsproblem.Alan Turing - 1936 - Proceedings of the London Mathematical Society 42 (1):230-265.
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  • Russell's Mathematical Logic.Kurt Gödel - 1944 - In The Philosophy of Bertrand Russell. Northwestern University Press. pp. 123-154.
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  • Introduction to mathematical logic.Elliott Mendelson - 1964 - Princeton, N.J.,: Van Nostrand.
    The Fourth Edition of this long-established text retains all the key features of the previous editions, covering the basic topics of a solid first course in ...
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  • Der wahrheitsbegriff in den formalisierten sprachen.Alfred Tarski - 1935 - Studia Philosophica 1:261--405.
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  • The concept of truth in formalized languages.Alfred Tarski - 1956 - In Logic, semantics, metamathematics. Oxford,: Clarendon Press. pp. 152--278.
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  • Remarks on the foundations of mathematics.Ludwig Wittgenstein - 1956 - Oxford [Eng.]: Blackwell. Edited by G. E. M. Anscombe, Rush Rhees & G. H. von Wright.
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  • Friedman's Research on Subsystems of Second Order Arithmetic.Stephen G. Simpson - 1990 - Journal of Symbolic Logic 55 (2):870-874.
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  • Nonprovability of Certain Combinatorial Properties of Finite Trees.Stephen G. Simpson - 1990 - Journal of Symbolic Logic 55 (2):868-869.
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  • Infinity and the mind: the science and philosophy of the infinite.Rudy von Bitter Rucker - 1982 - Princeton, N.J.: Princeton University Press.
    In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Here Rucker acquaints us with Gödel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of infinity, he (...)
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  • Computability & unsolvability.Martin Davis - 1958 - New York: Dover Publications.
    Classic text considersgeneral theory of computability, computable functions, operations on computable functions, Turing machines self-applied, unsolvable decision problems, applications of general theory, mathematical logic, Kleene hierarchy, computable functionals, classification of unsolvable decision problems and more.
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  • Inductive Full Satisfaction Classes.Henryk Kotlarski & Zygmunt Ratajczyk - 1990 - Annals of Pure and Applied Logic 47 (1):199--223.
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  • Hilbert's program and the omega-rule.Aleksandar Ignjatović - 1994 - Journal of Symbolic Logic 59 (1):322 - 343.
    In the first part of this paper we discuss some aspects of Detlefsen's attempt to save Hilbert's Program from the consequences of Godel's Second Incompleteness Theorem. His arguments are based on his interpretation of the long standing and well-known controversy on what, exactly, finitistic means are. In his paper [1] Detlefsen takes the position that there is a form of the ω-rule which is a finitistically valid means of proof, sufficient to prove the consistency of elementary number theory Z. On (...)
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  • Countable algebra and set existence axioms.Harvey M. Friedman - 1983 - Annals of Pure and Applied Logic 25 (2):141.
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  • To and from philosophy — discussions with gödel and Wittgenstein.Hao Wang - 1991 - Synthese 88 (2):229 - 277.
    I propose to sketch my views on several aspects of the philosophy of mathematics that I take to be especially relevant to philosophy as a whole. The relevance of my discussion would, I think, become more evident, if the reader keeps in mind the function of (the philosophy of) mathematics in philosophy in providing us with more transparent aspects of general issues. I shall consider: (1) three familiar examples; (2) logic and our conceptual frame; (3) communal agreement and objective certainty; (...)
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  • Zur Widerspruchsfreiheit der Zahlentheorie.Wilhelm Ackermann - 1940 - Journal of Symbolic Logic 5 (3):125-127.
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  • The axiomatization of arithmetic.Hao Wang - 1957 - Journal of Symbolic Logic 22 (2):145-158.
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  • Finitism.W. W. Tait - 1981 - Journal of Philosophy 78 (9):524-546.
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  • Partial realizations of Hilbert's program.Stephen G. Simpson - 1988 - Journal of Symbolic Logic 53 (2):349-363.
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  • Hilbert's program sixty years later.Wilfried Sieg - 1988 - Journal of Symbolic Logic 53 (2):338-348.
    On June 4, 1925, Hilbert delivered an address to the Westphalian Mathematical Society in Miinster; that was, as a quick calculation will convince you, almost exactly sixty years ago. The address was published in 1926 under the title Über dasUnendlicheand is perhaps Hilbert's most comprehensive presentation of his ideas concerning the finitist justification of classical mathematics and the role his proof theory was to play in it. But what has become of the ambitious program for securing all of mathematics, once (...)
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  • Solution of a problem of Leon Henkin.M. H. Löb - 1955 - Journal of Symbolic Logic 20 (2):115-118.
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  • Hilbert's epistemology.Philip Kitcher - 1976 - Philosophy of Science 43 (1):99-115.
    Hilbert's program attempts to show that our mathematical knowledge can be certain because we are able to know for certain the truths of elementary arithmetic. I argue that, in the absence of a theory of mathematical truth, Hilbert does not have a complete theory of our arithmetical knowledge. Further, while his deployment of a Kantian notion of intuition seems to promise an answer to scepticism, there is no way to complete Hilbert's epistemology which would answer to his avowed aims.
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  • A remark concerning decidability of complete theories.Antoni Janiczak - 1950 - Journal of Symbolic Logic 15 (4):277-279.
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  • Wittgenstein's philosophy of mathematics.Michael Dummett - 1959 - Philosophical Review 68 (3):324-348.
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  • On an alleged refutation of Hilbert's program using gödel's first incompleteness theorem.Michael Detlefsen - 1990 - Journal of Philosophical Logic 19 (4):343 - 377.
    It is argued that an instrumentalist notion of proof such as that represented in Hilbert's viewpoint is not obligated to satisfy the conservation condition that is generally regarded as a constraint on Hilbert's Program. A more reasonable soundness condition is then considered and shown not to be counter-exemplified by Godel's First Theorem. Finally, attention is given to the question of what a theory is; whether it should be seen as a "list" or corpus of beliefs, or as a method for (...)
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  • A note on the entscheidungsproblem.Alonzo Church - 1936 - Journal of Symbolic Logic 1 (1):40-41.
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  • Godel's theorem, church's theorem, and mechanism.J. J. C. Smart - 1961 - Synthese 13 (1):105-10.
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  • The emperor’s new mind.Roger Penrose - 1989 - Oxford University Press.
    Winner of the Wolf Prize for his contribution to our understanding of the universe, Penrose takes on the question of whether artificial intelligence will ever ...
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  • Godel's theorem is a red Herring.I. J. Good - 1968 - British Journal for the Philosophy of Science 19 (February):357-8.
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  • Lucas' number is finally up.G. Lee Bowie - 1982 - Journal of Philosophical Logic 11 (3):279-85.
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  • Computing machinery and intelligence.Alan M. Turing - 1950 - Mind 59 (October):433-60.
    I propose to consider the question, "Can machines think?" This should begin with definitions of the meaning of the terms "machine" and "think." The definitions might be framed so as to reflect so far as possible the normal use of the words, but this attitude is dangerous, If the meaning of the words "machine" and "think" are to be found by examining how they are commonly used it is difficult to escape the conclusion that the meaning and the answer to (...)
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  • Der erste Widerspruchsfreiheitsbeweis für die klassische Zahlentheorie.Gerhard Gentzen - 1974 - Archive for Mathematical Logic 16 (3-4):97-118.
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  • Remarks on the Foundations of Mathematics.Ludwig Wittgenstein - 1956 - Oxford: Macmillan. Edited by G. E. M. Anscombe, Rush Rhees & G. H. von Wright.
    Wittgenstein's work remains, undeniably, now, that off one of those few philosophers who will be read by all future generations.
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