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EB

ONLINE

Cham : Springer International Publishing AG, 2021

1 online resource (274 pages)

ISBN 9783030658243 (electronic bk.)

ISBN 9783030658236

Logic, Epistemology, and the Unity of Science Ser. ; v.51

Print version: Heinzmann, Gerhard Paul Lorenzen -- Mathematician and Logician Cham : Springer International Publishing AG,c2021 ISBN 9783030658236

Intro -- Preface -- Contents -- List of Contributors -- 1 Paul Lorenzens Weg von der Mathematik zur Philosophie - Personliche Erinnerungen -- 2 Operation and Predicativity: Lorenzen’s Approach to Arithmetic -- 1 Introduction -- 2 The main philosophical considerations -- 3 The construction of the natural and the real numbers -- 4 A short outlook on predicative mathematics -- References -- 3 Conceptions of Infinity and Set in Lorenzen’s Operationist System -- 1 Introduction -- 2 Operationism and the foundational crisis -- 3 Elimination of the classical notion of set -- 4 The question of infinity -- 4.1 A shift in focus -- 4.2 Constructs and infinity -- 4.3 Rejection of actual infinity -- 5 Conclusion and outlook -- References -- 4 Lorenzen and Constructive Mathematics -- Introduction -- 1 Lorenzen’s analysis of Gentzen’s work -- 1.1 The consistency proof -- 1.2 The inversion principle -- 1.3 Distributive lattices and entailment relations -- 2 Proof-theoretic analysis of point-free spaces -- 3 Measure theory -- 3.1 Borel subsets of Cantor space -- 3.2 Borel’s measure problem -- 3.3 An inductive solution of Borel’s measure problem -- 4 Game semantics -- References -- 5 Lorenzen between Gentzen and Schutte -- 1 Introduction: Hilbert’s Programme after Gentzen -- 2 Lorenzen’s consistency proof for ramified type theory (without reducibility) -- 3 Gentzen, Bernays, Schutte -- 4 Digression: Lorenzen’s 1951 Zeitschrift paper -- 5 Conclusion -- References -- 6 Syntax for Semantics: Krull’s Maximal Ideal Theorem -- 1 Introduction -- 2 Maximal ideals -- 3 Entailment relations -- 4 Krull’s theorem without choice -- 5 Applications -- Acknowledgements -- References -- 7 Regular Entailment Relations -- Introduction -- 1 General properties of regular entailment relations -- 2 Another presentation of regular entailment relations -- 3 Equivariant systems of ideals.

10 The correspondence between Hasse and Lorenzen, 1953-1963 -- 11 The correspondence between Aubert and Lorenzen, 1978-1979 -- References.

4 Regularisation of an equivariant system of ideals -- 5 A constructive version of the Lorenzen-Clifford-Dieudonne Theorem -- 6 Prufer’s definition of the regularisation -- 7 The l-group structure in the noncommutative case -- 8 Examples -- Acknowledgements -- References -- 8 Connecting Sequent Calculi with Lorenzen-Style Dialogue Games -- 1 Introduction -- 2 Some sequent calculi -- 3 Two different information extraction games -- 4 Relating games and calculi -- 5 Lorenzen-style games? -- 6 Game variants for other substructural calculi -- 7 Conclusion - an extended research agenda -- References -- 9 Lorenzen’s Reshaping of Krull’s Fundamentalsatz for Integral Domains (1938-1953) -- 1 Introduction -- 2 Krull 1930: a first attempt at introducing valuations for an integral domain -- 3 Krull 1932: the Fundamentalsatz for integral domains -- 4 Krull 1936a: the computational content of the Fundamentalsatz -- 5 Lorenzen 1939: the Fundamentalsatz for preordered cancellative monoids -- 5.1 Systems of ideals -- 5.2 Ideals in a lattice-preordered group -- 5.3 Transfer to the system of t-ideals -- 6 Lorenzen 1950: the Fundamentalsatz without valuations -- 7 Lorenzen 1952: the Fundamentalsatz for semilattice domains -- 8 Lorenzen 1953: the Fundamentalsatz for integral domains as an embedding into a super-l-group -- 9 A letter from Krull to Scholz from 1953: the well-ordering theorem -- References -- 10 Lorenzen’s Correspondence with Hasse, Krull, and Aubert, Together with Some Relevant Documents -- 1 Synopsis -- 2 The correspondence between Krull and Lorenzen, 1938 -- 3 The reports on Lorenzen’s thesis -- 4 The correspondence between Hasse and Lorenzen, 1938-1942 -- 6 The correspondence between Krull and Lorenzen, 1943-1944 -- 7 A postcard from Lorenzen to Hasse, 1945 -- 8 Documents relating to Lorenzen’s career, 1945-1946 -- 9 A letter from Krull to Scholz, 1953.

001895903

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(Au-PeEL)EBL6707454

(MiAaPQ)EBC6707454

(OCoLC)1265405286