Paul Lorenzen – Mathematician and Logician

Description

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