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Paul Lorenzen -- Mathematician and Logician.

Bibliographic Details
Main Author: Heinzmann, Gerhard.
Other Authors: Wolters, Gereon.
Format: eBook
Language:English
Published: Cham : Springer International Publishing AG, 2021.
Edition:1st ed.
Series:Logic, Epistemology, and the Unity of Science Series
Subjects:
Electronic books.
Online Access:Click to View
Table of Contents:
  • Intro
  • Preface
  • Contents
  • List of Contributors
  • 1 Paul Lorenzens Weg von der Mathematik zur Philosophie - Persönliche 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 Schütte
  • 1 Introduction: Hilbert's Programme after Gentzen
  • 2 Lorenzen's consistency proof for ramified type theory (without reducibility)
  • 3 Gentzen, Bernays, Schütte
  • 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.
  • 4 Regularisation of an equivariant system of ideals
  • 5 A constructive version of the Lorenzen-Clifford-Dieudonné Theorem
  • 6 Prüfer'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.
  • 10 The correspondence between Hasse and Lorenzen, 1953-1963
  • 11 The correspondence between Aubert and Lorenzen, 1978-1979
  • References.

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