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

by Ferreiros Dominguez, Jose

• ISBN13: 9783764357498
• ISBN10: 3764357495
• Format: Hardcover
• Copyright: 1999-11-01
• Publisher: Springer Verlag

Summary

"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)

Table of Contents

Introduction	xi
Aims and Scope	xiii
General Historiographical Remarks	xvii
Part One: The Emergence of Sets within Mathematics	1	(168)
Institutional and Intellectual Contexts in German Mathematics, 1800--1870	3	(36)
Mathematics at the Reformed German Universities	4	(6)
Traditional and `Modern' Foundational Viewpoints	10	(8)
The Issue of the Infinite	18	(6)
The Gottingen Group, 1855--1859	24	(8)
The Berlin School, 1855--1870	32	(7)
A New Fundamental Notion: Riemann's Manifolds	39	(42)
The Historical Context: Grossenlehre, Gauss, and Herbart	41	(6)
Logical Prerequisites	47	(6)
The Mathematical Context of Riemann's Innovation	53	(9)
Riemann's General Definition	62	(5)
Manifolds, Arithmetic, and Topology	67	(3)
Riemann's Influence on the Development of Set Theory	70	(11)
Appendix: Riemann and Dedekind	77	(4)
Dedekind and the Set-theoretical Approach to Algebra	81	(36)
The Algebraic Origins of Dedekind's Set Theory, 1856--58	82	(8)
A New Fundamental Notion for Algebra: Fields	90	(4)
The Emergence of Algebraic Number Theory	94	(5)
Ideals and Methodology	99	(8)
Dedekind's Infinitism	107	(4)
The Diffusion of Dedekind's Views	111	(6)
The Real Number System	117	(28)
`Construction' vs. Axiomatization	119	(5)
The Definitions of the Real Numbers	124	(11)
The Influence of Riemann: Continuity in Arithmetic and Geometry	135	(2)
Elements of the Topology of R	137	(8)
Origins of the Theory of Point-Sets	145	(24)
Dirichlet and Riemann: Transformations in the Theory of Real Functions	147	(7)
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	(4)
Crystallization of the Notion of Content	165	(4)
Part Two: Entering the Labyrinth -- Toward Abstract Set Theory	169	(128)
The Notion of Cardinality and the Continuum Hypothesis	171	(44)
The Relations and Correspondence Between Cantor and Dedekind	172	(4)
Non-denumerability of R	176	(7)
Cantor's Exposition and the `Berlin Circumstances'	183	(4)
Equipollence of Continua R and Rn	187	(10)
Cantor's Difficulties	197	(5)
Derived Sets and Cardinalities	202	(6)
Cantor's Definition of the Continuum	208	(2)
Further Efforts on the Continuum Hypothesis	210	(5)
Sets and Maps as a Foundation for Mathematics	215	(42)
Origins of Dedekind's Program for the Foundations of Arithmetic	218	(6)
Theory of Sets, Mappings, and Chains	224	(8)
Through the Natural Numbers to Pure Mathematics	232	(7)
Dedekind and the Cantor--Bernstein Theorem	239	(2)
Dedekind's Theorem of Infinity, and Epistemology	241	(7)
Reception of Dedekind's Ideas	248	(9)
The Transfinite Ordinals and Cantor's Mature Theory	257	(40)
``Free Mathematics''	259	(4)
Cantor's Notion of Set in the Early 1880s	263	(4)
The Transfinite (Ordinal) Numbers	267	(7)
Ordered Sets	274	(8)
The Reception in the Early 1880s	282	(4)
Cantor's Theorem	286	(2)
The Beitrage zur Begrundung der transfiniten Mengenlehre	288	(2)
Cantor and the Paradoxes	290	(7)
Part Three: In Search of an Axiom System	297	(96)
Diffusion, Crisis, and Bifurcation: 1890 to 1914	299	(38)
Spreading Set Theory	300	(6)
The Complex Emergence of the Paradoxes	306	(5)
The Axiom of Choice and the Early Foundational Debate	311	(6)
The Early Work of Zermelo	317	(8)
Russell's Theory of Types	325	(8)
Other Developments in Set Theory	333	(4)
Logic and Type Theory in the Interwar Period	337	(28)
An Atmosphere of Insecurity: Weyl, Brouwer, Hilbert	338	(7)
Diverging Conceptions of Logic	345	(3)
The Road to the Simple Theory of Types	348	(5)
Type Theory at its Zenith	353	(4)
A Radical Proposal: Weyl and Skolem on First-Order Logic	357	(8)
Consolidation of Axiomatic Set Theory	365	(28)
The Contributions of Fraenkel	366	(4)
Toward the Modern Axiom System: von Neumann and Zermelo	370	(8)
The System von Neumann-Bernays-Godel	378	(4)
Godel's Relative Consistency Results	382	(4)
First-Order Axiomatic Set Theory	386	(2)
A Glance Ahead: Mathematicians and Foundations after World War II	388	(5)
Bibliographical References	393	(29)
Index of Illustrations	422	(1)
Name Index	423	(7)
Subject Index	430