Paul Lorenzen -- Mathematician and Logician.

• Superior document: Logic, Epistemology, and the Unity of Science Series ; v.51
• Author: Heinzmann, Gerhard.
• TeilnehmendeR: Wolters, Gereon.
• Place / Publishing House: Cham : : Springer International Publishing AG,, 2021.; ©2021.
• Year of Publication: 2021
• Edition: 1st ed.
• Language: English
• Series: Logic, Epistemology, and the Unity of Science Series
• Physical Description: 1 online resource (274 pages)

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.