Roads to infinity: the mathematics of truth and proof
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Natick, MA
Peters
[2010]
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index A K Peters Book gehört zu CRC Press / Taylor & Francis Group |
| Physical Description: | xi, 203 Seiten Illustrationen 24 cm |
| ISBN: | 9781568814667 1568814666 |
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MARC
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Record in the Search Index
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| adam_text | Titel: Roads to infinity
Autor: Stillwell, John
Jahr: 2010
Contents
Preface ix
1 The Diagonal Argument 1
1.1 Counting and Countability................. 2
1.2 Does One Infinite Size Fit All? ............... 4
1.3 Cantor s Diagonal Argument................ 6
1.4 Transcendental Numbers................... 10
1.5 Other Uncountability Proofs................. 12
1.6 Rates of Growth........................ 14
1.7 The Cardinality of the Continuum............. 16
1.8 Historical Background.................... 19
2 Ordinals 29
2.1 Counting Past Infinity.................... 30
2.2 The Countable Ordinals................... 33
2.3 The Axiom of Choice..................... 37
2.4 The Continuum Hypothesis................. 40
2.5 Induction............................ 42
2.6 Cantor Normal Form..................... 46
2.7 Goodstein s Theorem..................... 47
2.8 Hercules and the Hydra................... 51
2.9 Historical Background.................... 54
3 COMPUTABILITY AND PROOF 67
3.1 Formal Systems........................ 68
3.2 Post s Approach to Incompleteness............. 72
3.3 Gödel s First Incompleteness Theorem........... 75
3.4 Gödel s Second Incompleteness Theorem......... 80
3.5 Formalization of Computability............... 82
3.6 The Halting Problem..................... 85
3.7 The Entscheidungsproblem................. 87
3.8 Historical Background.................... 89
vin Contents
4 Logic 97
4.1 Prepositional Logic...................... 98
4.2 A Classical System...................... 100
4.3 A Cut-Free System for Propositional Logic........ 102
4.4 Happy Endings........................ 105
4.5 Predicate Logic........................ 106
4.6 Completeness, Consistency, Happy Endings....... 110
4.7 Historical Background.................... 112
5 Arithmetic 119
5.1 How Might We Prove Consistency? ............ 120
5.2 Formal Arithmetic ...................... 121
5.3 The Systems PA and ÑÁù.................. 122
5.4 Embedding PA in PAo,.................... 124
5.5 Cut Elimination in ÑÁù ................... 127
5.6 The Height of This Great Argument............ 130
5.7 Roads to Infinity ....................... 133
5.8 Historical Background.................... 135
6 Natural Unprovable Sentences 139
6.1 A Generalized Goodstein Theorem............. 140
6.2 Countable Ordinals via Natural Numbers......... 141
6.3 From Generalized Goodstein to Well-Ordering...... 144
6.4 Generalized and Ordinary Goodstein........... 146
6.5 Provably Computable Functions.............. 147
6.6 Complete Disorder Is Impossible.............. 151
6.7 The Hardest Theorem in Graph Theory.......... 154
6.8 Historical Background.................... 157
7 Axioms of Infinity 165
7.1 Set Theory without Infinity................. 165
7.2 Inaccessible Cardinals.................... 168
7.3 The Axiom of Determinacy................. 170
7.4 Largeness Axioms for Arithmetic.............. 172
7.5 Large Cardinals and Finite Mathematics.......... 173
7.6 Historical Background.................... 177
Bibliography 183
Index 189
|
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| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.3/22 |
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| discipline | Mathematik Philosophie |
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| spelling | Stillwell, John 1942- Verfasser (DE-588)128427264 aut Roads to infinity the mathematics of truth and proof John Stillwell Natick, MA Peters [2010] © 2010 xi, 203 Seiten Illustrationen 24 cm txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index A K Peters Book gehört zu CRC Press / Taylor & Francis Group Set theory Infinite Logic, Symbolic and mathematical Oneindigheid gtt Verzamelingen (wiskunde) gtt Symbolische logica gtt Beweistheorie (DE-588)4145177-6 gnd rswk-swf Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Unendlichkeit (DE-588)4136067-9 gnd rswk-swf Unendlichkeit (DE-588)4136067-9 s Beweistheorie (DE-588)4145177-6 s Mathematische Logik (DE-588)4037951-6 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022490588&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Stillwell, John 1942- Roads to infinity the mathematics of truth and proof Set theory Infinite Logic, Symbolic and mathematical Oneindigheid gtt Verzamelingen (wiskunde) gtt Symbolische logica gtt Beweistheorie (DE-588)4145177-6 gnd Mathematische Logik (DE-588)4037951-6 gnd Unendlichkeit (DE-588)4136067-9 gnd |
| subject_GND | (DE-588)4145177-6 (DE-588)4037951-6 (DE-588)4136067-9 |
| title | Roads to infinity the mathematics of truth and proof |
| title_auth | Roads to infinity the mathematics of truth and proof |
| title_exact_search | Roads to infinity the mathematics of truth and proof |
| title_full | Roads to infinity the mathematics of truth and proof John Stillwell |
| title_fullStr | Roads to infinity the mathematics of truth and proof John Stillwell |
| title_full_unstemmed | Roads to infinity the mathematics of truth and proof John Stillwell |
| title_short | Roads to infinity |
| title_sort | roads to infinity the mathematics of truth and proof |
| title_sub | the mathematics of truth and proof |
| topic | Set theory Infinite Logic, Symbolic and mathematical Oneindigheid gtt Verzamelingen (wiskunde) gtt Symbolische logica gtt Beweistheorie (DE-588)4145177-6 gnd Mathematische Logik (DE-588)4037951-6 gnd Unendlichkeit (DE-588)4136067-9 gnd |
| topic_facet | Set theory Infinite Logic, Symbolic and mathematical Oneindigheid Verzamelingen (wiskunde) Symbolische logica Beweistheorie Mathematische Logik Unendlichkeit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022490588&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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