Labyrinth of thought: a history of set theory and its role in modern mathematics
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
Basel [u.a.]
Birkhäuser
2007
|
| Ausgabe: | 2., rev. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturangaben |
| Beschreibung: | XXV, 466 S. Ill., graph. Darst. |
| ISBN: | 3764383496 9783764383497 |
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| adam_text | LABYRINTH OF THOUGHT A HISTORY OF SET THEORY AND ITS ROLE IN MODERN
MATHEMATICS JOSE FERREIROS SECOND REVISED EDITION BIRKHAUSER
BASEL-BOSTON-BERLIN CONTENTS INTRODUCTION XV 1. AIMS AND SCOPE XVII
2.GENERAL HISTORIOGRAPHICAL REMARKS XXI PART ONE: THE EMERGENCE OF SETS
WITHIN MATHEMATICS 1 I INSTITUTIONAL AND INTELLECTUAL 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.THEG6TTINGEN GROUP, 1855-1859 24 5.THE
BERLIN SCHOOL, 1855-1870 32 II A NEW FUNDAMENTAL NOTION: RIEMANN 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 APPENDIX: RIEMANN AND DEDEKIND 77 III DEDEKIND AND THE
SET-THEORETICAL APPROACH TO ALGEBRA 8 1 L.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 XII
CONTENTS 4. IDEALS AND METHODOLOGY 99 5.DEDEKIND S INFINITISM 107 6.THE
DIFFUSION OF DEDEKIND S VIEWS ; 1 11 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 M 137 V ORIGINS OF THE
THEORY OF POINT-SETS 145 L.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 L.THE RELATIONS AND CORRESPONDENCE BETWEEN CANTOR AND DEDEKIND 172
2.NON-DENUMERABILITY OF US. 176 3.CANTOR S EXPOSITION AND THE BERLIN
CIRCUMSTANCES 183 4.EQUIPOLLENCE OF CONTINUA 08. AND 08 187 5.CANTOR S
DIFFICULTIES 197 6.DERIVED SETS AND CARDINALITIES 202 7. CANTOR S
DEFINITION OF THE CONTINUUM T.. 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 CONTENTS . XIII 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
TRANSFINITE ORDINALS AND CANTOR S MATURE THEORY 257 L. TREEMATHEMATICS
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 THE EARLY
1880S 282 6.CANTOR S THEOREM 286 7. THE BEITRDGE ZUR BEGRIINDUNG DER
TRANSFINITEN MENGENLEHRE 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 ZERMELP 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 T.. 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 L.THE CONTRIBUTIONS OF
FRAENKEL 366 XIV CONTENTS 2. TOWARD THE MODERN AXIOM SYSTEM: VON NEUMANN
AND ZERMELO 370 3.THE SYSTEM VON NEUMANN-BERNAYS-GODEL 378 4.G6DEL S
RELATIVE CONSISTENCY RESULTS 382 5. FIRST-ORDER AXIOMATIC SET THEORY 386
6. A GLANCE AHEAD: MATHEMATICIANS AND FOUNDATIONS AFTER WORLD WAR II 388
BIBLIOGRAPHICAL REFERENCES 393 INDEX OF ILLUSTRATIONS 422 NAME INDEX 423
SUBJECT INDEX 430 EPILOGUE 2007 441
|
| adam_txt |
LABYRINTH OF THOUGHT A HISTORY OF SET THEORY AND ITS ROLE IN MODERN
MATHEMATICS JOSE FERREIROS SECOND REVISED EDITION BIRKHAUSER
BASEL-BOSTON-BERLIN CONTENTS INTRODUCTION XV 1. AIMS AND SCOPE XVII
2.GENERAL HISTORIOGRAPHICAL REMARKS XXI PART ONE: THE EMERGENCE OF SETS
WITHIN MATHEMATICS 1 I INSTITUTIONAL AND INTELLECTUAL 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.THEG6TTINGEN GROUP, 1855-1859 24 5.THE
BERLIN SCHOOL, 1855-1870 32 II A NEW FUNDAMENTAL NOTION: RIEMANN'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 APPENDIX: RIEMANN AND DEDEKIND 77 III DEDEKIND AND THE
SET-THEORETICAL APPROACH TO ALGEBRA 8 1 L.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 XII
CONTENTS 4. IDEALS AND METHODOLOGY 99 5.DEDEKIND'S INFINITISM 107 6.THE
DIFFUSION OF DEDEKIND'S VIEWS ; 1 11 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 M 137 V ORIGINS OF THE
THEORY OF POINT-SETS 145 L.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 L.THE RELATIONS AND CORRESPONDENCE BETWEEN CANTOR AND DEDEKIND 172
2.NON-DENUMERABILITY OF US. 176 3.CANTOR'S EXPOSITION AND THE 'BERLIN
CIRCUMSTANCES' 183 4.EQUIPOLLENCE OF CONTINUA 08. AND 08" 187 5.CANTOR'S
DIFFICULTIES 197 6.DERIVED SETS AND CARDINALITIES 202 7. CANTOR'S
DEFINITION OF THE CONTINUUM T. 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 CONTENTS . XIII 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
TRANSFINITE ORDINALS AND CANTOR'S MATURE THEORY 257 L.'TREEMATHEMATICS"
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 THE EARLY
1880S 282 6.CANTOR'S THEOREM 286 7. THE BEITRDGE ZUR BEGRIINDUNG DER
TRANSFINITEN MENGENLEHRE 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 ZERMELP 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 T. 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 L.THE CONTRIBUTIONS OF
FRAENKEL 366 XIV CONTENTS 2. TOWARD THE MODERN AXIOM SYSTEM: VON NEUMANN
AND ZERMELO 370 3.THE SYSTEM VON NEUMANN-BERNAYS-GODEL 378 4.G6DEL'S
RELATIVE CONSISTENCY RESULTS 382 5. FIRST-ORDER AXIOMATIC SET THEORY 386
6. A GLANCE AHEAD: MATHEMATICIANS AND FOUNDATIONS AFTER WORLD WAR II 388
BIBLIOGRAPHICAL REFERENCES 393 INDEX OF ILLUSTRATIONS 422 NAME INDEX 423
SUBJECT INDEX 430 EPILOGUE 2007 441 |
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| index_date | 2024-07-02T18:57:43Z |
| indexdate | 2024-07-09T21:08:07Z |
| institution | BVB |
| isbn | 3764383496 9783764383497 |
| language | German |
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| spelling | Ferreirós Domínguez, José Verfasser (DE-588)121021351 aut Labyrinth of thought a history of set theory and its role in modern mathematics José Ferreirós 2., rev. ed. Basel [u.a.] Birkhäuser 2007 XXV, 466 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturangaben Geschichte 1850-1940 gnd rswk-swf Geschichte Set theory History Geschichte (DE-588)4020517-4 gnd rswk-swf Mengenlehre (DE-588)4074715-3 gnd rswk-swf Mengenlehre (DE-588)4074715-3 s Geschichte (DE-588)4020517-4 s DE-604 Geschichte 1850-1940 z HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016144184&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ferreirós Domínguez, José Labyrinth of thought a history of set theory and its role in modern mathematics Geschichte Set theory History Geschichte (DE-588)4020517-4 gnd Mengenlehre (DE-588)4074715-3 gnd |
| subject_GND | (DE-588)4020517-4 (DE-588)4074715-3 |
| title | Labyrinth of thought a history of set theory and its role in modern mathematics |
| title_auth | Labyrinth of thought a history of set theory and its role in modern mathematics |
| title_exact_search | Labyrinth of thought a history of set theory and its role in modern mathematics |
| title_exact_search_txtP | Labyrinth of thought a history of set theory and its role in modern mathematics |
| title_full | Labyrinth of thought a history of set theory and its role in modern mathematics José Ferreirós |
| title_fullStr | Labyrinth of thought a history of set theory and its role in modern mathematics José Ferreirós |
| title_full_unstemmed | Labyrinth of thought a history of set theory and its role in modern mathematics José Ferreirós |
| title_short | Labyrinth of thought |
| title_sort | labyrinth of thought a history of set theory and its role in modern mathematics |
| title_sub | a history of set theory and its role in modern mathematics |
| topic | Geschichte Set theory History Geschichte (DE-588)4020517-4 gnd Mengenlehre (DE-588)4074715-3 gnd |
| topic_facet | Geschichte Set theory History Mengenlehre |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016144184&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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