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Hilbert’s Program Then and Now
, 2005
"... Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and els ..."
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Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s
Hilbert’s “Verunglückter Beweis,” the first epsilon theorem and consistency proofs. History and Philosophy of Logic
"... Abstract. On the face of it, Hilbert’s Program was concerned with proving consistency of mathematical systems in a finitary way. This was to be accomplished by showing that that these systems are conservative over finitistically interpretable and obviously sound quantifierfree subsystems. One propo ..."
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Abstract. On the face of it, Hilbert’s Program was concerned with proving consistency of mathematical systems in a finitary way. This was to be accomplished by showing that that these systems are conservative over finitistically interpretable and obviously sound quantifierfree subsystems. One proposed method of giving such proofs is Hilbert’s epsilonsubstitution method. There was, however, a second approach which was not refelected in the publications of the Hilbert school in the 1920s, and which is a direct precursor of Hilbert’s first epsilon theorem and a certain “general consistency result. ” An analysis of this socalled “failed proof ” lends further support to an interpretation of Hilbert according to which he was expressly concerned with conservatitvity proofs, even though his publications only mention consistency as the main question. §1. Introduction. The aim of Hilbert’s program for consistency proofs in the 1920s is well known: to formalize mathematics, and to give finitistic consistency proofs of these systems and thus to put mathematics on a “secure foundation.” What is perhaps less well known is exactly how Hilbert thought this should be carried out. Over ten years before Gentzen developed sequent calculus formalizations
Remarks On Finitism
 Reflections on the Foundations of Mathematics. Essays in Honor of Solomon Feferman, LNL 15. Association for Symbolic Logic
, 2000
"... representability in intuition. (See [2, p. 40].) But our problem is, of course, not the finiteness of a number, but the infinity of numbers. There is, I think, a di#culty with Bernays' notion of formal object, where this is intended to extend to numbers so large as, not only to be beyond proces ..."
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representability in intuition. (See [2, p. 40].) But our problem is, of course, not the finiteness of a number, but the infinity of numbers. There is, I think, a di#culty with Bernays' notion of formal object, where this is intended to extend to numbers so large as, not only to be beyond processing by the human mind, but possibly to be beyond representablity in the physical world. [2, p. 39]. This di#culty ought to be discussed more adequately then + This paper is based on a talk that I was very pleased to give at the conference Reflections, December 1315, 1998, in honor of Solomon Feferman on his seventieth birthday. The choice of topic is especially appropriate for the conference in view of recent discussions we had had about finitism. I profited from the discussion following my talk and, in particular, from the remarks of Richard Zach. I have since had the advantage of further discussions with Zach and of reading his paper 1998; and I use his scholarshi
Gödel on Intuition and on Hilbert’s finitism
"... There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the con ..."
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There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, Gödel’s writings represent a smooth evolution, with just one rather small doublereversal, of his view of finitism. He used the term “finit ” (in German) or “finitary ” or “finitistic ” primarily to refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [Gödel, 1938a] and the lecture notes for a lecture at Yale [Gödel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of firstorder number theory, P A; but starting in the Dialectica paper
Numbers And Functions In Hilbert's Finitism
, 1998
"... or concrete? Some of the most fruitful sources on the topic of Hilbert's conception of finitism are his 1922 and 1926 papers, his collaborator Bernays's exchange with Mller (Mller 1923, Bernays 1923), as well as the relevant sections in Hilbert and Bernays (1934, 1939). In 1905, Hilbert gi ..."
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or concrete? Some of the most fruitful sources on the topic of Hilbert's conception of finitism are his 1922 and 1926 papers, his collaborator Bernays's exchange with Mller (Mller 1923, Bernays 1923), as well as the relevant sections in Hilbert and Bernays (1934, 1939). In 1905, Hilbert gives a first account of finitistic number theory in terms of strokes and equality signs. We note here that no identification of certain (sequences of) signs with numbers is made, rather, the sequences of 1's and ='s are divided into two classes, the class of entities (these are the sequences of the form "1...1 = 1...1" with equal numbers of 1's on the left and right) and the class of nonentities; the former are the true propositions. Hence we have here a finitistic account, not of numbers, but of numerical truth.
The Termite and the Tower: Goodstein sequences and provability in PA
, 2007
"... We discuss Goodstein’s Theorem, a true finitary statement that can only be proven by infinitary means. We assume very little knowledge of logic and provide the necessary background to understand both Goodstein’s Theorem and, at a high level, how to show that a statement is true but not provable ins ..."
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We discuss Goodstein’s Theorem, a true finitary statement that can only be proven by infinitary means. We assume very little knowledge of logic and provide the necessary background to understand both Goodstein’s Theorem and, at a high level, how to show that a statement is true but not provable inside a given set of axioms. 1
Reconstructing Hilbert to Construct Category Theoretic Structuralism
"... This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, its relation to the “algebraic”1 approach to mathematical structuralism. I first consider the FregeHilbert debate with the aim of distinguishing b ..."
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This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, its relation to the “algebraic”1 approach to mathematical structuralism. I first consider the FregeHilbert debate with the aim of distinguishing between axioms as assertions, i.e., as statements that are used to express or assert truths about a unique subject matter, and an axiom system as a schema that is used to provide “a system of conditions for what might be called a relational structure ” (Bernays [1967], p. 497) so that axioms, as implicit definitions, are about whatever satisfies the conditions set forth. I then use this inquiry to reevaluate arguments against using category theory to frame an algebraic structuralist philosophy of mathematics. Hellman has argued that category theory cannot stand on its own as a “foundation ” for a structuralist interpretation of mathematics because “the problem of the home address remains ” (Hellman [2003], pgs. 8 & 15). That is, since the axioms for a category “merely tell us what it is to be a structure of a certain kind ” and because “its axioms are not assertory ” (Ibid. 7), we need a background mathematical theory whose axioms are