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The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program
, 2001
"... . After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the ..."
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. After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for ever stronger and more comprehensive areas of mathematics and finitistic proofs of consistency of these systems. Early advances in these areas were made by Hilbert (and Bernays) in a series of lecture courses at the University of Gttingen between 1917 and 1923, and notably in Ackermann 's dissertation of 1924. The main innovation was the invention of the ecalculus, on which Hilbert's axiom systems were based, and the development of the esubstitution method as a basis for consistency proofs. The paper traces the development of the "simultaneous development of logic and mathematics" through the enotation and provides an analysis of Ackermann's consisten...
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, 2007
"... responsible for all errors that remain. Unless otherwise indicated in the text, translations from German are by the author. The original German passages will be confined to footnotes wherever possible. 1 1 ..."
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responsible for all errors that remain. Unless otherwise indicated in the text, translations from German are by the author. The original German passages will be confined to footnotes wherever possible. 1 1
The Gödel Paradox and Wittgenstein’s Reasons 1. The Implausibile Wittgenstein Wittgenstein’s notorious comments on Gödel’s First Incompleteness Theorem in the
"... Remarks on the Foundations of Mathematics were dismissed by early commentators, such as Kreisel, Anderson, Dummett, and Bernays, as an unfortunate episode in the career of a great philosopher. It appears that Wittgenstein had in his sights only the informal account of the Theorem, presented by Gödel ..."
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Remarks on the Foundations of Mathematics were dismissed by early commentators, such as Kreisel, Anderson, Dummett, and Bernays, as an unfortunate episode in the career of a great philosopher. It appears that Wittgenstein had in his sights only the informal account of the Theorem, presented by Gödel in the introduction of his celebrated 1931 paper, and was misguided by it (not that he was the only one: because of the misunderstandings it originated, Helmer said that that exposition “without any claim to complete precision”1 is the only mistake in Gödel’s paper). It is claimed that Wittgenstein erroneously considered essential the natural language interpretation of the Gödel sentence, whose undecidability within (the modified system considered by Gödel, taken from) Russell and Whitehead’s Principia mathematica is at the core of the First Theorem, as claiming “I am not provable”. On the contrary, Gödel’s proof can be framed in syntactic terms in which no extramathematical interpretation of the formulas is required. Commentators were particularly struck by the fact that Wittgenstein seems to take the Gödel formula as a paradoxical sentence, not too different from the usual Liar – and Gödel’s proof itself, therefore, as the deduction of an inconsistency: 11. Let us suppose I prove the unprovability (in Russell’s system) of P; then by this proof I have proved P. Now if this proof were one in Russell’s system – I should in this case have proved at once that it belonged and did not belong to Russell’s system. – That is what comes of making up such
Wittgenstein’s comments on Gödel’s First Incompleteness Theorem
"... An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard dis ..."
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An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match with some intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question.