Labyrinth of Thought

by Ferreiros, Jose

• ISBN13: 9783764383497
• ISBN10: 3764383496
• Edition: 2nd
• Format: Paperback
• Copyright: 2007-09-03
• Publisher: Birkhauser

Summary

Labyrinth of Thought discusses the emergence and development of set theory and the set-theoretic approach to mathematics during the period 1850-1940. Rather than focusing on the pivotal figure of Georg Cantor, it analyzes his work and the emergence of transfinite set theory within the broader context of the rise of modern mathematics. The text has a tripartite structure. Part 1, The Emergence of Sets within Mathematics, surveys the initial motivations for a mathematical notion of a set within several branches of the discipline (geometry, algebra, algebraic number theory, real and complex analysis), emphasizing the role played by Riemann in fostering acceptance of the set-theoretic approach. In Part 2, Entering the Labyrinth, attention turns to the earliest theories of sets, their evolution, and their reception by the mathematical community; prominent are the epoch-making contributions of Cantor and Dedekind, and the complex interactions between them. Part 3, In Search of an Axiom System, studies the four-decade period from the discovery of set-theoretic paradoxes to Gödel's independence results, an era during which set theory gradually became assimilated into mainstream mathematics; particular attention is given to the interactions between axiomatic set theory and modern systems of formal logic, especially the interplay between set theory and type theory. A new Epilogue for this second edition offers further reflections on the foundations of set theory, including the "dichotomy conception" and the well-known iterative conception.

Table of Contents

Introduction	p. xv
Aims and Scope	p. xvii
General Historiographical Remarks	p. xxi
The Emergence of Sets within Mathematics	p. 1
Institutional and Intellectual Contexts in German Mathematics, 1800-1870	p. 3
Mathematics at the Reformed German Universities	p. 4
Traditional and 'Modern' Foundational Viewpoints	p. 10
The Issue of the Infinite	p. 18
The Gottingen Group, 1855-1859	p. 24
The Berlin School, 1855-1870	p. 32
A New Fundamental Notion: Riemann's Manifolds	p. 39
The Historical Context: Grossenlehre, Gauss, and Herbart	p. 41
Logical Prerequisites	p. 47
The Mathematical Context of Riemann's Innovation	p. 53
Riemann's General Definition	p. 62
Manifolds, Arithmetic, and Topology	p. 67
Riemann's Influence on the Development of Set Theory	p. 70
Riemann and Dedekind	p. 77
Dedekind and the Set-theoretical Approach to Algebra	p. 81
The Algebraic Origins of Dedekind's Set Theory, 1856-58	p. 82
A New Fundamental Notion for Algebra: Fields	p. 90
The Emergence of Algebraic Number Theory	p. 94
Ideals and Methodology	p. 99
Dedekind's Infinitism	p. 107
The Diffusion of Dedekind's Views	p. 111
The Real Number System	p. 117
'Construction' vs. Axiomatization	p. 119
The Definitions of the Real Numbers	p. 124
The influence of Riemann: Continuity in Arithmetic and Geometry	p. 135
Elements of the Topology of R	p. 137
Origins of the Theory of Point-Sets	p. 145
Dirichlet and Riemann: Transformations in the Theory of Real Functions	p. 147
Lipschitz and Hankel on Nowhere Dense Sets and Integration	p. 154
Cantor on Sets of the First Species	p. 157
Nowhere Dense Sets of the Second Species	p. 161
Crystallization of the Notion of Content	p. 165
Entering the Labyrinth - Toward Abstract Set Theory	p. 169
The Notion of Cardinality and the Continuum Hypothesis	p. 171
The Relations and Correspondence Between Cantor and Dedekind	p. 172
Non-denumerability of R	p. 176
Cantor's Exposition and the 'Berlin Circumstances'	p. 183
Equipollence of Continua R and R[superscript n]	p. 187
Cantor's Difficulties	p. 197
Derived Sets and Cardinalities	p. 202
Cantor's Definition of the Continuum	p. 208
Further Efforts on the Continuum Hypothesis	p. 210
Sets and Maps as a Foundation for Mathematics	p. 215
Origins of Dedekind's Program for the Foundations of Arithmetic	p. 218
Theory of Sets, Mappings, and Chains	p. 224
Through the Natural Numbers to Pure Mathematics	p. 232
Dedekind and the Cantor-Bernstein Theorem	p. 239
Dedekind's Theorem of Infinity, and Epistemology	p. 241
Reception of Dedekind's Ideas	p. 248
The Transfinite Ordinals and Cantor's Mature Theory	p. 257
"Free Mathematics"	p. 259
Cantor's Notion of Set in the Early 1880s	p. 263
The Transfinite (Ordinal) Numbers	p. 267
Ordered Sets	p. 274
The Reception in the Early 1880s	p. 282
Cantor's Theorem	p. 286
The Beitrage zur Begrundung der transfiniten Mengenlehre	p. 288
Cantor and the Paradoxes	p. 290
In Search of an Axiom System	p. 297
Diffusion, Crisis, and Bifurcation: 1890 to 1914	p. 299
Spreading Set Theory	p. 300
The Complex Emergence of the Paradoxes	p. 306
The Axiom of Choice and the Early Foundational Debate	p. 311
The Early Work of Zermelo	p. 317
Russell's Theory of Types	p. 325
Other Developments in Set Theory	p. 333
Logic and Type Theory in the Interwar Period	p. 337
An Atmosphere of Insecurity: Weyl, Brouwer, Hilbert	p. 338
Diverging Conceptions of Logic	p. 345
The Road to the Simple Theory of Types	p. 348
Type Theory at its Zenith	p. 353
A Radical Proposal: Weyl and Skolem on First-Order Logic	p. 357
Consolidation of Axiomatic Set Theory	p. 365
The Contributions of Fraenkel	p. 366
Toward the Modern Axiom System: von Neumann and Zermelo	p. 370
The System von Neumann-Bernays-Godel	p. 378
Godel's Relative Consistency Results	p. 382
First-Order Axiomatic Set Theory	p. 386
A Glance Ahead: Mathematicians and Foundations after World War II	p. 388
Bibliographical References	p. 393
Index of Illustrations	p. 422
Name Index	p. 423
Subject Index	p. 430
Epilogue 2007	p. 441
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