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  1. 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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  • Recursive predicates and quantifiers.S. C. Kleene - 1943 - Transactions of the American Mathematical Society 53:41-73.
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  • Arithmetization of Metamathematics in a General Setting.Solomon Feferman - 1960 - Journal of Symbolic Logic 31 (2):269-270.
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  • Die Widerspruchsfreiheit der reinen Zahlentheorie.Gerhard Gentzen - 1936 - Journal of Symbolic Logic 1 (2):75-75.
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  • On the Elementary Theory of Linear Order.J. Leonard - 1968 - Journal of Symbolic Logic 33 (2):287-287.
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  • Formally Self-Referential Propositions for Cut Free Analysis and Related Systems.Georg Kreisel & Gaisi Takeuti - 1985 - Journal of Symbolic Logic 50 (1):244-246.
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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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  • Zur Widerspruchsfreiheit der Zahlentheorie.Wilhelm Ackermann - 1940 - Journal of Symbolic Logic 5 (3):125-127.
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  • The Elementary Theory of Finite Fields.James Ax - 1973 - Journal of Symbolic Logic 38 (1):162-163.
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  • Undecidable-Rings.Raphael M. Robinson - 1952 - Journal of Symbolic Logic 17 (4):268-269.
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  • Was Sind und was Sollen Die Zahlen?Richard Dedekind - 1888 - Cambridge University Press.
    This influential 1888 publication explained the real numbers, and their construction and properties, from first principles.
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  • Philosophie der Arithmetik.E. S. Husserl - 1892 - Philosophical Review 1 (3):327-330.
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  • (1 other version)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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  • (1 other version)Inductive Full Satisfaction Classes.Henryk Kotlarski & Zygmunt Ratajczyk - 1990 - Annals of Pure and Applied Logic 47 (1):199--223.
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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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  • An Argument against the Plausibility of Church's Thesis.László Kalmár - 1959 - In A. Heyting (ed.), Constructivity in mathematics: Proceedings of the colloquium held at Amsterdam 1957. Amsterdam: North-Holland. pp. 72-80.
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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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  • 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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  • Thirty years of foundational studies.Andrzej Mostowski - 1966 - New York,: Barnes & Noble.
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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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  • The unprovability of consistency: an essay in modal logic.George Boolos - 1979 - New York: Cambridge University Press.
    The Unprovability of Consistency is concerned with connections between two branches of logic: proof theory and modal logic. Modal logic is the study of the principles that govern the concepts of necessity and possibility; proof theory is, in part, the study of those that govern provability and consistency. In this book, George Boolos looks at the principles of provability from the standpoint of modal logic. In doing so, he provides two perspectives on a debate in modal logic that has persisted (...)
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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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  • 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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  • Finitism.W. W. Tait - 1981 - Journal of Philosophy 78 (9):524-546.
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  • Which set existence axioms are needed to prove the cauchy/peano theorem for ordinary differential equations?Stephen G. Simpson - 1984 - Journal of Symbolic Logic 49 (3):783-802.
    We investigate the provability or nonprovability of certain ordinary mathematical theorems within certain weak subsystems of second order arithmetic. Specifically, we consider the Cauchy/Peano existence theorem for solutions of ordinary differential equations, in the context of the formal system RCA 0 whose principal axioms are ▵ 0 1 comprehension and Σ 0 1 induction. Our main result is that, over RCA 0 , the Cauchy/Peano Theorem is provably equivalent to weak Konig's lemma, i.e. the statement that every infinite {0, 1}-tree (...)
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  • Definability and decision problems in arithmetic.Julia Robinson - 1949 - Journal of Symbolic Logic 14 (2):98-114.
    In this paper, we are concerned with the arithmetical definability of certain notions of integers and rationals in terms of other notions. The results derived will be applied to obtain a negative solution of corresponding decision problems.In Section 1, we show that addition of positive integers can be defined arithmetically in terms of multiplication and the unary operation of successorS(whereSa=a+ 1). Also, it is shown that both addition and multiplication can be defined arithmetically in terms of successor and the relation (...)
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  • On the philosophical significance of consistency proofs.Michael D. Resnik - 1974 - Journal of Philosophical Logic 3 (1/2):133 - 147.
    We have seen that despite Feferman's results Gödel's second theorem vitiates the use of Hilbert-type epistemological programs and consistency proofs as a response to mathematical skepticism. Thus consistency proofs fail to have the philosophical significance often attributed to them.This does not mean that consistency proofs are of no interest to philosophers. We know that a ‘non-pathological’ consistency proof for a system S will use methods which are not available in S. When S is as strong a system as we are (...)
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  • A survey of proof theory.G. Kreisel - 1968 - Journal of Symbolic Logic 33 (3):321-388.
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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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  • Systems of predicative analysis.Solomon Feferman - 1964 - Journal of Symbolic Logic 29 (1):1-30.
    This paper is divided into two parts. Part I provides a resumé of the evolution of the notion of predicativity. Part II describes our own work on the subject.Part I§1. Conceptions of sets.Statements about sets lie at the heart of most modern attempts to systematize all (or, at least, all known) mathematics. Technical and philosophical discussions concerning such systematizations and the underlying conceptions have thus occupied a considerable portion of the literature on the foundations of mathematics.
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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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  • (1 other version)A note on the entscheidungsproblem.Alonzo Church - 1936 - Journal of Symbolic Logic 1 (1):40-41.
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  • Shadows of the Mind: A Search for the Missing Science of Consciousness.Roger Penrose - 1994 - Oxford University Press.
    Presenting a look at the human mind's capacity while criticizing artificial intelligence, the author makes suggestions about classical and quantum physics and ..
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  • (1 other version)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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  • (1 other version)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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  • (1 other version)A Decision Method for Elementary Algebra and Geometry.Alfred Tarski - 1949 - Journal of Symbolic Logic 14 (3):188-188.
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  • (2 other versions)Satan Stultified.J. R. Lucas - 1968 - The Monist 52 (1):145-158.
    The application of Gödel’s theorem to the problem of minds and machines is difficult. Paul Benacerraf makes the entirely valid ‘Duhemian’ point that the argument is not, and cannot be, a purely mathematical one, but needs some philosophical premisses to be able to yield any philosophical conclusions. Moreover, the philosophical premisses are of very different kinds. Some are concerned with what is essential to being a machine—these are typically intricate, but definite, easily formalised by the mathematician, but unintelligible to the (...)
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  • (1 other version)Second Thoughts about Church's Thesis and Mathematical Proofs.Elliott Mendelson - 1990 - Journal of Philosophy 87 (5):225-233.
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  • (2 other versions)Self-Reference and Modal Logic.George Boolos & C. Smorynski - 1988 - Journal of Symbolic Logic 53 (1):306.
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  • Primitive Recursive Functions.Raphael M. Robinson - 1948 - Journal of Symbolic Logic 13 (2):113-114.
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  • Formal Reductions of the General Combinatorial Decision Problem.Emil L. Post - 1943 - Journal of Symbolic Logic 8 (1):50-52.
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