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Computational Content of Classical Logic
 SEMANTICS AND LOGICS OF COMPUTATION
, 1996
"... This course is an introduction to the research trying to connect the proof theory of classical logic and computer science. We omit important and standard topics, among them the connection between the computational interpretation of classical logic and the programming operator callcc. Instead, here ..."
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This course is an introduction to the research trying to connect the proof theory of classical logic and computer science. We omit important and standard topics, among them the connection between the computational interpretation of classical logic and the programming operator callcc. Instead, here we put the emphasis on actual mathematical examples. We analyse the following questions: what can be the meaning of a noneoeective proof of an existential statement, a statement that claims the existence of a nite object that satises a decidable property? Is it clear that a noneoeective proof has a meaning at all? Can we always say that this proof contains implicitly, if not explicitly, some eoeective witness? Is this witness unique? By putting the emphasis on actual mathematical examples, we follow Gentzen who founded natural deduction by analysing concrete mathematical examples, like Euclid's proof of the innity of prime numbers. We
Program development by proof transformation
"... We begin by reviewing the natural deduction rules for the!^8fragment of minimal logic. It is shown how intuitionistic and classical logic can be embedded. Recursion and induction is added to obtain a more realistic proof system. Simple types are added in order to make the language more expressive. ..."
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We begin by reviewing the natural deduction rules for the!^8fragment of minimal logic. It is shown how intuitionistic and classical logic can be embedded. Recursion and induction is added to obtain a more realistic proof system. Simple types are added in order to make the language more expressive. We also consider two alternative methods to deal with the strong or constructive existential quantifier 9\Lambda. Finally we discuss the wellknown notion of an extracted program of a derivation involving 9\Lambda, in order to set up a relation between the two alternatives. Section 2 deals with the computational content of classical proofs. As is wellknown a proof of a 89theorem with a quantifierfree kernel where 9 is viewed as defined by:8: can be used as a program. We describe a "direct method " to use such a proof as a program, and compare it with Harvey Friedman's Atranslation [3] followed by program extraction from the resulting constructive proof. It is shown that both algorithms coincide. In section 3 Goad's method of pruning of proof trees is introduced. It is shown how a proof can be simplified after addition of some further assumptions. In a first step some subproofs are replaced by different ones using the additional assumptions. In a second step parts of the proof tree are pruned, i.e. cut out. Note that the first step involves searching for new proofs using the new assumptions of formulas in the proof tree. Hence we also have to discuss proof search in minimal logic. Finally section 4 treats an example already considered by Goad in his thesis [5], the binpacking problem. The main difference to Goad's work is that he used a logic with the strong existential quantifier, whereas we work within the!8fragment. This example is particularly wellsuited to demonstrate that the pruning method can be applied to adapt programs to particular situations, and moreover that pruning can change the functions computed by programs. In this sense this method is essentially different from program development by program transformation. We would like to thank Michael Bopp and KarlHeinz Niggl for their help in preparing these notes.
"Clarifying the Nature of the Infinite": the development of metamathematics and proof theory
, 2001
"... We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic 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 ..."
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We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic 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.
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
Arguments for the Continuity Principle
, 2000
"... Contents 1 The continuity principle 1 2 A phenomenological consideration 8 2.1 An argument for G(raph)WCN . . . . . . . . . . . . . . . . . 8 2.2 Two arguments against WCN . . . . . . . . . . . . . . . . . . 13 3 Other arguments for continuity 15 3.1 Undecidability of equality of choice sequences ..."
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Contents 1 The continuity principle 1 2 A phenomenological consideration 8 2.1 An argument for G(raph)WCN . . . . . . . . . . . . . . . . . 8 2.2 Two arguments against WCN . . . . . . . . . . . . . . . . . . 13 3 Other arguments for continuity 15 3.1 Undecidability of equality of choice sequences . . . . . . . . . 15 3.2 Kripke's Schema and full PEM . . . . . . . . . . . . . . . . . 15 3.3 The KLST theorem . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Conclusion 19 1 The continuity principle There are two principles that lend Brouwer's mathematics the extra power beyond arithmetic. Both are presented in Brouwer's writings with little or no argument. One, the principle of bar induction, will not concern us here. The other, the continuity principle for numbers, occurs for the rst time in print in [Brouwer 1918]. It is formulated and immediately applied to show that the set of numerical choice sequences is not enumerable. In fa
Metapredicative Subsystems of Analysis
 Ph.D. thesis, Institut für Informatik und angewandte Mathematik, Univeristät Bern, 2000. & EXPLICIT MAHLO 21
, 2001
"... In this paper we present some metapredicative subsystems of analysis. We deal with reflection principles, #model existence axioms (limit axioms) and axioms asserting the existence of hierarchies. We show several equivalences of the introduced subsystems. In particular we prove the equivalence of # ..."
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In this paper we present some metapredicative subsystems of analysis. We deal with reflection principles, #model existence axioms (limit axioms) and axioms asserting the existence of hierarchies. We show several equivalences of the introduced subsystems. In particular we prove the equivalence of # 1 1 transfinite dependent choice and # 1 2 reflection on #models of # 1 1 DC. 1 Introduction The formal system of classical analysis is second order arithmetic with the full comprehension principle. It was baptized classical analysis, since classical mathematical analysis can be formalized in it. Often, subsystems of classical analysis su#ce as formal framework for particular parts of mathematical analysis. During the last decades a lot of such subsystems have been isolated and prooftheoretically investigated. The subsystems of analysis introduced in this paper belong to metapredicative prooftheory. Metapredicative systems have prooftheoretic ordinals beyond # 0 but can still be tr...
Hermann Weyl’s Intuitionistic Mathematics. Dirk
"... It is common knowledge that for a short while Hermann Weyl joined Brouwer in his pursuit of a revision of mathematics according to intuitionistic principles. There is, however, little in the literature that sheds light on Weyl’s role, and in particular on Brouwer’s reaction to Weyl’s allegiance to t ..."
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It is common knowledge that for a short while Hermann Weyl joined Brouwer in his pursuit of a revision of mathematics according to intuitionistic principles. There is, however, little in the literature that sheds light on Weyl’s role, and in particular on Brouwer’s reaction to Weyl’s allegiance to the cause of intuitionism. This short episode certainly raises a number of questions: what made Weyl give up his own program, spelled out in “Das Kontinuum”, how come Weyl was so wellinformed about Brouwer’s new intuitionism, in what respect did Weyl’s intuitionism differ from Brouwer’s intuitionism, what did Brouwer think of Weyl’s views,........? To some of these questions at least partial answers can be put forward on the basis of some of the available correspondence and notes. The present paper will concentrate mostly on the historical issues of the intuitionistic episode in Weyl’s career. Weyl entered the foundational controversy with a bang in 1920 with his sensational paper “On the new foundational crisis in mathematics ” 1. He had already made a name for himself in the foundations of mathematics in 1918 with his monograph “The Continuum” [Weyl 1918] ; this contained in addition to a technical logical – mathematical construction of the continuum, a fairly extensive discussion of the shortcomings of the traditional construction of the continuum on the basis of arbitrary — and hence also impredicative — Dedekind cuts. This book did not cause much of a stir in mathematics, that is to say, it was ritually quoted in the literature but, probably, little understood. It had to wait for a proper appreciation until the phenomenon of impredicativity was better understood 2. The paper “On the new foundational crisis in mathematics ” had a totally different effect, it was the proverbial stone thrown into the quiet pond of mathematics. Weyl characterised it in retrospect with the somewhat apologetic words: Only with some hesitation I acknowledge these lectures, which reflect in their style, which was here and there really bombastic, the mood of excited times — the times immediately following the First World War. 3 Indeed, Weyl’s “New crisis ” reads as a manifesto to the mathematical community, it uses an evocative language with a good many explicit references to the political
JACQUES HERBRAND: LIFE, LOGIC, AND AUTOMATED DEDUCTION
"... The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1 ..."
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The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.
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
The irreflexivity of Brouwer’s philosophy ∗
, 2000
"... I argue that Brouwer’s general philosophy cannot account for itself, and, a fortiori, cannot lend justification to mathematical principles derived from it. Thus it cannot ground intuitionism, the job Brouwer had intended it to do. The strategy is to ask whether that philosophy actually allows for th ..."
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I argue that Brouwer’s general philosophy cannot account for itself, and, a fortiori, cannot lend justification to mathematical principles derived from it. Thus it cannot ground intuitionism, the job Brouwer had intended it to do. The strategy is to ask whether that philosophy actually allows for the kind of knowledge that such an account of itself would amount to. Brouwer tried to go ‘from philosophy to mathematics ’ and grounded his intuitionistic mathematics in a more general philosophy. 1 This background philosophy can be characterized as a transcendental one. That is, it purports to explain how a nonmundane subject builds up its world in consciousness. It is a radical transcendental philosophy in that this ‘world ’ does not contain just physical objects but everything, including abstract objects and the mundane subject (the subject as part of the world). From the empirical point of view, such a nonmundane subject is an idealized one. Like fellow transcendentalists