The Development of Mathematical Logic from Russell to Tarski: 1900-1935

Leila Haaparanta, ed., The History of Modern Logic. New York and Oxford: Oxford University Press, 2009, pp. 318-471 (with Paolo Mancosu and Calixto Badesa)

Reprinted in Paolo Mancosu, The Adventure of Reason. Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940. Oxford: Oxford University press, 2010

The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight “itineraries” concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski’s work on truth.

DOI: 10.1093/acprof:oso/9780195137316.003.0029


Table of Contents

Introduction 1
Itinerary I: Metatheoretical Properties of Axiomatic Systems 3
1.1 Introduction 3
1.2 Peano’s school on the logical structure of theories 4
1.3 Hilbert on axiomatization 8
1.4 Completeness and categoricity in the work of Veblen and Huntington 10
1.5 Truth in a structure 12
Itinerary II: Bertrand Russell’s Mathematical Logic 15
2.1 From the Paris congress to the Principles of Mathematics 1900–1903 15
2.2 Russell and Poincare on predicativity 19
2.3 On Denoting 21
2.4 Russell’s ramified type theory 22
2.5 The logic of Principia 25
2.6 Further developments 26
Itinerary III: Zermelo’s Axiomatization of Set Theory and Related Foundational Issues 29
3.1 The debate on the axiom of choice 29
3.2 Zermelo’s axiomatization of set theory 32
3.3 The discussion on the notion of “definit” 35
3.4 Metatheoretical studies of Zermelo’s axiomatization 38
Itinerary IV: The Theory of Relatives and Löwenheim’s Theorem 41
4.1 Theory of relatives and model theory 41
4.2 The logic of relatives 44
4.3 Löwenheim’s theorem 46
4.4 Skolem’s first versions of Löwenheim’s theorem 56
Itinerary V: Logic in the Hilbert School 59
5.1 Early lectures on logic 59
5.2 The completeness of propositional logic 60
5.3 Consistency and completeness 61
5.4 Axioms and inference rules 66
5.5 Grundzüge der theoretischen Logik 70
5.6 The decision problem 71
5.6.1 The decision problem in the tradition of algebra of logic 72
5.6.2 Work on the decision problem after 1920 73
5.7 Combinatory logic and ?-calculus 74
5.8 Structural inference: Hertz and Gentzen 76
Itinerary VI: Proof Theory and Arithmetic 81
6.1 Hilbert’s Program for consistency proofs 81
6.2 Consistency proofs for weak fragments of arithmetic 82
6.3 Ackermann and von Neumann on epsilon substitution 87
6.4 Herbrand’s Theorem 92
6.5 Kurt Gödel and the incompleteness theorems 94
Itinerary VII: Intuitionism and Many-valued Logics 99
7.1 Intuitionistic logic 99
7.1.1 Brouwer’s philosophy of mathematics 99
7.1.2 Brouwer on the excluded middle 101
7.1.3 The logic of negation 102
7.1.4 Kolmogorov 103
7.1.5 The debate on intuitionist logic 106
7.1.6 The formalization and interpretation of intuitionistic logic 108
7.1.7 Gödel’s contributions to the metatheory of intuitionistic logic 110
7.2 Many-valued logics 111
Itinerary VIII: Semantics and Model-theoretic Notions 117
8.1 Background 117
8.1.1 The algebra of logic tradition 117
8.1.2 Terminological variations (systems of objects, models, and structures) 118
8.1.3 Interpretations for propositional logic 119
8.2 Consistency and independence for propositional logic 120
8.3 Post’s contributions to the metatheory of the propositional calculus 123
8.4 Semantical completeness of first-order logic 124
8.5 Models of first order logic 129
8.6 Completeness and categoricity 130
8.7 Tarski’s definition of truth 134
Notes 141
Bibliography 149
Index of Citations 175


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