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

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430 | |

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Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics Jose Ferreiros No preview available - 2001 |