. Introduction, xi
1: Aims and Scope, xiii
2: General Historical Remarks, xvii
. Part One: The Emergence of Sets within Mathematics, 1
. I Institutional and Intellectural Contexts in German Mathematics, 1800-1870, 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, 1855-1859, 24
5: The Berlin School, 1855-1870, 32
. II A New Fundamental Notion: Reimann's Manifolds, 39
1: The Historical Context: Grossenlehre, Gauss, and Herbart, 41
2: Logical Prerequisites, 47
3:The Mathematical Context of Riemann's Innovation, 53
4: Riemann's General Definition, 62
5: Manifolds, Arithmetic, and Topology, 67
6 Riemann's Influence on the Development of Set Theory, 70
<i>Appendix</i>: Riemann and Dedekind, 77
. III Dedekind and the Set-Theoretical Approach to Algebra, 81
1: The Algebraic Origins of Dedekind's Set Theory, 1856-58, 82
2: A New Fundamental Notion for Algebra: Fields, 90
3: The Emergence of Algebraic Number Theory, 94
4: Ideals and Methodology, 99
5: Dedekind's Infinitism, 107
6: The Diffusion of Dedekind's Views, 111
. IV The Real Number System, 117
1: 'Construction' vs. Axiomatization, 119
2: The Definitions of the Real Numbers, 124
3: The Influence of Riemann: Continuity in Arithmetic and Geometry, 135
4: Elements of the Topology of <b>R</b>, 137
. V Origins of the Theory of Point-Sets, 145
1: Dirichlet and Riemann: 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
. Part Two: Entering the Labyrinth - Toward Abstract Set Theory, 169
. VI The Notion of Cardinality and the Continuum Hypothesis, 171
1: The relations and Corespondence Between Cantor and Dedekind, 172
2: Non-denumerability of <b>R</b>, 176
3: Cantor's Exposition and the 'Berlin Circumstances', 183
4: Equipollence of Continua <b>R</b> and <b>R</b><sup><i>n</i></sup>, 187
5: Cantor's Difficulties, 197
6: Derived Sets and Cardinalities, 202
7: Cantor's Definition of the Continuum, 208
8: Further Efforts on the Continuum Hypothesis, 210
. VII Sets and Maps as A Foundation for Mathematics, 215
1: Origins of Dedekind's 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 Cantor-Bernstein Theorem, 239
5: Dedekind's Theorem of Infinity, and Epistemology, 241
6: Reception of Dedekind's Ideas, 248
. VIII The Transfinit Ordinals and Cantor's Mature Theory, 257
1: "Free Mathematics", 259
2: Cantor's Notion of Set in the Early 1880s, 263
3: The Transfinite (Ordinal) Numbers, 267
4: Ordered Sets, 274
5: The Reception in Early 1880s, 282
6: Cantor's Theorem, 286
7: The <i>Beitrage zur Begründung der transfiniten Mengenlehre</i>, 288
8: Cantor and the Paradoxes, 290
. Part Three: In Search of an Axiom System, 297
. IX 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: Russell's Theory of Types, 325
6: Other Developments in Set Theory, 333
. X Logic and Type Theory in the Interwar Period, 337
1: An atmosphere of Insecurity: 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
5: A Radical Proposal: Weyl and Skolem on First-Order Logic, 357
. XI Consolidation of Axiomatic Set Theory, 365
1: The Contributions of Fraenkel, 336
2: Toward the Modern Axiom System: von Neumann and Zermelo, 370
3: The System von Neumann-Bernays-Gödel, 378
4: Gödel's Relative Consistency Results, 382
5: First-Order Axiomatic Set Theory, 386
6: A Glance Ahead: Mathematics and Foundations after World War II, 388
. Bibliographical References, 393
. Index of Illustrations, 422
. Name Index, 423
. Subject Index, 430