Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics

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Springer Science & Business Media, Nov 1, 2001 - Mathematics - 440 pages
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"José Ferreirós has written a magisterial account of the history of set theory which is panoramic, balanced, and engaging. Not only does this book synthesize much previous work and provide fresh insights and points of view, but it also features a major innovation, a full-fledged treatment of the emergence of the set-theoretic approach in mathematics from the early nineteenth century. This takes up Part One of the book. Part Two analyzes the crucial developments in the last quarter of the nineteenth century, above all the work of Cantor, but also Dedekind and the interaction between the two. Lastly, Part Three details the development of set theory up to 1950, taking account of foundational questions and the emergence of the modern axiomatization." (Bulletin of Symbolic Logic)

 

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Contents

Institutional and Intellectual Contexts in German Mathematics 18001870
3
1 Mathematics at the Reformed German Universities
4
2 Traditional and Modern Foundational Viewpoints
10
3 The Issue of the Infinite
18
4 The Göttingen Group 18551859
24
5 The Berlin School 18551870
32
A New Fundamental Notion Riemanns Manifolds
39
Grössenlehre Gauss and Herbart
41
Sets and Maps as a Foundation for Mathematics
215
1 Origins of Dedekinds Program for the Foundations of Arithmetic
218
2 Theory of Sets Mappings and Chains
224
3 Through the Natural Numbers to Pure Mathematics
232
4 Dedekind and the CantorBernstein Theorem
239
5 Dedekinds Theorem of Infinity and Epistemology
241
6 Reception of Dedekinds ldeas
248
The Transfinite Ordinals and Cantors Mature Theory
257

2 Logical Prerequisites
47
3 The Mathematical Context of Riemanns Innovation
53
4 Riemanns General Definition
62
5 Manifolds Arithmetic and Topology
67
6 Riemanns Influence on the Development of Set Theory
70
Riemann and Dedekind
77
Dedekind and the Settheoretical Approach to Algebra
81
1 The Algebraic Origins of Dedekinds Set Theory 185658
82
Fields
90
3 The Emergence of Algebraic Number Theory
94
4 Ideals and Methodology
99
5 Dedekinds Infinitism
107
6 The Diffusion of Dedekinds Views
111
The Real Number System
117
1 Construction vs Axiomatization
119
2 The Definitions of the Real Numbers
124
Continuity in Arithmetic and Geometry
135
4 Elements of the Topology of ℝ
137
Origins of the Theory of PointSets
145
Transformations in the Theory of Real Functions
147
2 Lipschitz and Hankel on Nowhere Dense Sets and Integration
154
3 Cantor on Sets of the First Species
157
4 Nowhere Dense Sets of the Second Species
161
5 Crystallization of the Notion of Content
165
Entering the Labyrinth Toward Abstract Set Theory
169
The Notion of Cardinality and the Continuum Hypothesis
171
1 The Relations and Correspondence Between Cantor and Dedekind
172
2 Nondenumerability of ℝ
177
3 Cantors Exposition and the Berlin Circumstances
183
4 Equipollence of Continua ℝ and ℝⁿ
187
5 Cantors Difficulties
197
6 Derived Sets and Cardinalities
202
7 Cantors Definition of the Continuum
208
8 Further Efforts on the Continuum Hypothesis
210
1 Free Mathematics
259
2 Cantors Notion of Set in the Early 1880s
263
3 The Transfinite Ordinal Numbers
267
4 Ordered Sets
274
5 The Reception in the Early 1880s
282
6 Cantors Theorem
286
7 The Beiträge zur Begründung der transfiniten Mengenlehre
288
8 Cantor and the Paradoxes
290
In Search of an Axiom System
297
Diffusion Crisis and Bifurcation 1890 to 1914
299
1 Spreading Set Theory
300
2 The Complex Emergence of the Paradoxes
306
3 The Axiom of Choice and the Early Foundational Debate
311
4 The Early Work of Zermelo
317
5 Russells Theory of Types
325
6 Other Developments in Set Theory
333
Logic and Type Theory in the Interwar Period
337
Weyl Brouwer Hilbert
338
2 Diverging Conceptions of Logic
345
3 The Road to the Simple Theory of Types
348
4 Type Theory at its Zenith
353
Weyl and Skolem on FirstOrder Logic
357
Consolidation of Axiomatic Set Theory
365
1 The Contributions of Fraenkel
366
von Neumann and Zermelo
370
3 The System von NeumannBernaysGödel
378
4 Gödels Relative Consistency Results
382
5 FirstOrder Axiomatic Set Theory
386
Mathematicians and Foundations after World War II
388
Bibliographical References
393
Index of Illustrations
422
Name Index
423
Subject Index
430
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