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History and Future of Implicit and Inductionless Induction: Beware the Old Jade and The Zombie!
, 2005
"... In this survey on implicit induction I recollect some memories on the history of implicit induction as it is relevant for future research on computerassisted theorem proving, esp. memories that significantly differ from the presentation in a recent handbook article on “inductionless induction”. M ..."
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In this survey on implicit induction I recollect some memories on the history of implicit induction as it is relevant for future research on computerassisted theorem proving, esp. memories that significantly differ from the presentation in a recent handbook article on “inductionless induction”. Moreover, the important references excluded there are provided here. In order to clear the fog a little, there is a short introduction to inductive theorem proving and a discussion of connotations of implicit induction like “descente infinie”, “inductionless induction”, “proof by consistency”, implicit induction orderings (term orderings), and refutational completeness.
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
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.
From Heyting's arithmetic to verified programs
, 1998
"... We discuss higher type constructions inherent to intuitionistic proofs. As an example we consider Gentzen's proof of transnite induction up to the ordinal 0 . From the constructive content of this proof we derive higher type algorithms for some ordinal recursive hierarchies of number theoretic funct ..."
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We discuss higher type constructions inherent to intuitionistic proofs. As an example we consider Gentzen's proof of transnite induction up to the ordinal 0 . From the constructive content of this proof we derive higher type algorithms for some ordinal recursive hierarchies of number theoretic functions.
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
mlq header will be provided by the publisher Classical truth in higher types
"... We study, from a classical point of view, how the truth of a statement about higher type functionals depends on the underlying model. The models considered are the classical settheoretic finite type hierarchy and the constructively more meaningful models of Continuous Functionals, Hereditarily Effe ..."
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We study, from a classical point of view, how the truth of a statement about higher type functionals depends on the underlying model. The models considered are the classical settheoretic finite type hierarchy and the constructively more meaningful models of Continuous Functionals, Hereditarily Effective Operations, as well as the closed term model of Gödel’s system T. The main results are characterisations of prenex classes for which truth in the full settheoretic model transfers to truth in the other models. As a corollary we obtain that the axiom of choice is not conservative over Gödel’s system T with classical logic for closed ∃ 2formulas. We hope that this study will contribute to a clearer delineation and perception of constructive mathematics from a classical perspective. Copyright line will be provided by the publisher
Ordinal Analyzes and Large Cardinals Abstract
, 2005
"... One of the major topics of proof theoretical research are ordinal analyzes of axiom system. Ordinal analyzes go back to Gentzen’s 1943 paper [1] in which he proved that the order–type of any elementarily definable ordering whose well–foundedness is provable from the axioms of Peano arithmetic has an ..."
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One of the major topics of proof theoretical research are ordinal analyzes of axiom system. Ordinal analyzes go back to Gentzen’s 1943 paper [1] in which he proved that the order–type of any elementarily definable ordering whose well–foundedness is provable from the axioms of Peano arithmetic has an order type less than ε0:= min { α ω α = α} while conversely every ordinal less than ε0 can represented by an elementarily definable ordering whose well–foundedness is provable from the Peano axioms. Since then we define the proof theoretical ordinal of a theory T as the supremum of the order–types of all elementarily definable orderings whose well–foundedness is provable from T. The computation of the proof theoretical ordinal of a theory T is called an ordinal analysis for T. There is a variety of alternative definitions of the proof theoretical ordinal of a theory. For theories in the language of set theory the proof theoretical ordinal of a theory T can be defined as the least ordinal α such that the αth level of the constructible hierarchy is closed under the ωCK 1 –recursive functions whose totality is provable in T. The proof theoretical ordinal of a (reasonable) theory is always a constructive ordinal