Labyrinth of Thought: A History of Set Theory and Its Role in Modern MathematicsLabyrinth 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. |
Contents
3 | |
II | 39 |
The Real Number System | 117 |
Origins of the Theory of PointSets | 145 |
Entering the Labyrinth | 169 |
Sets and Maps as a Foundation for Mathematics | 215 |
Through the Natural Numbers to Pure Mathematics | 232 |
Dedekind and the CantorBernstein Theorem | 239 |
1890 to 1914 | 299 |
Spreading Set Theory | 300 |
The Complex Emergence of the Paradoxes | 306 |
The Axiom of Choice and the Early Foundational Debate | 311 |
The Early Work of Zermelo | 317 |
Russells Theory of Types | 325 |
Other Developments in Set Theory | 333 |
Logic and Type Theory in the Interwar Period | 337 |
Dedekinds Theorem of Infinity and Epistemology | 241 |
Reception of Dedekinds Ideas | 248 |
The Transfinite Ordinals and Cantors Mature Theory | 257 |
Free Mathematics | 259 |
Cantors Notion of Set in the Early 1880s | 263 |
The Transfinite Ordinal Numbers | 267 |
Ordered Sets | 274 |
The Reception in the Early 1880s | 282 |
Cantors Theorem | 286 |
The Beiträge zur Begründung der transfiniten Mengenlehre | 288 |
Cantor and the Paradoxes | 290 |
In Search of an Axiom System | 297 |
Weyl Brouwer Hilbert | 338 |
Diverging Conceptions of Logic | 345 |
The Road to the Simple Theory of Types | 348 |
Type Theory at its Zenith | 353 |
Weyl and Skolem on FirstOrder Logic | 357 |
Consolidation of Axiomatic Set Theory | 365 |
The Contributions of Fraenkel | 366 |
Bibliographical References | 393 |
422 | |
430 | |
Epilogue 2007 | 441 |
Other editions - View all
Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics José Ferreirós No preview available - 2007 |