Inconsistent mathematics:
The theory of inconsistency has been growing steadily over the last two decades. One focus has been philosophical issues arising from the paradoxes of set theory and semantics. A second focus has been the study of paraconsistent, or inconsistency-tolerant, logics. A third focus has been the applicat...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Dordrecht u.a.
Kluwer
1995
|
| Schriftenreihe: | Mathematics and its applications
312 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | The theory of inconsistency has been growing steadily over the last two decades. One focus has been philosophical issues arising from the paradoxes of set theory and semantics. A second focus has been the study of paraconsistent, or inconsistency-tolerant, logics. A third focus has been the application of paraconsistent logics to problems in artificial intelligence. This book focuses on a fourth aspect, the construction of mathematical theories in which contradictions occur, and the investigation of their properties. The inconsistent approach provides a distinctive perspective on the various number systems, order, differential and integral calculus, discontinuous changes, inconsistent systems of linear equations, projective geometry, topology and category theory. The final chapter outlines several known results concerning paradoxes in the foundations of set theory and semantics The book begins with an informal chapter which summarises the main results nontechnically, and draws philosophical implications from them. This volume will be of interest of advanced undergraduates, graduate students and professionals in the areas of logic, philosophy, mathematics and theoretical computer science |
| Beschreibung: | IX, 155 S. |
| ISBN: | 0792331869 |
Internformat
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| 490 | 1 | |a Mathematics and its applications |v 312 | |
| 520 | 3 | |a The theory of inconsistency has been growing steadily over the last two decades. One focus has been philosophical issues arising from the paradoxes of set theory and semantics. A second focus has been the study of paraconsistent, or inconsistency-tolerant, logics. A third focus has been the application of paraconsistent logics to problems in artificial intelligence. This book focuses on a fourth aspect, the construction of mathematical theories in which contradictions occur, and the investigation of their properties. The inconsistent approach provides a distinctive perspective on the various number systems, order, differential and integral calculus, discontinuous changes, inconsistent systems of linear equations, projective geometry, topology and category theory. The final chapter outlines several known results concerning paradoxes in the foundations of set theory and semantics | |
| 520 | |a The book begins with an informal chapter which summarises the main results nontechnically, and draws philosophical implications from them. This volume will be of interest of advanced undergraduates, graduate students and professionals in the areas of logic, philosophy, mathematics and theoretical computer science | ||
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Datensatz im Suchindex
| _version_ | 1804124512580534272 |
|---|---|
| adam_text | TABLE OF CONTENTS
ONE MOTIVATIONS
1. Paraconsistency 1
2. Summary 5
3. Philosophical Implications 7
TWO ARITHMETIC
1. Relevant Arithmetic 15
2. Nonclassical Logics and their Theories 20
3. Models for Number Systems with
Arithmetical Operations 27
4. Summary of Further Results in Arithmetic 31
THREE MODULO INFINITY
1. The Classical Denumerable Nonstandard Model
of Natural Number Arithmetic 33
2. Inconsistency 35
FOUR ORDER
1. Order and Equality without Function Symbols 39
2. Order and Equality with Function Symbols 40
FIVE CALCULUS
1. Introduction 43
2. A Nilpotent Ring of Hyperreal Numbers 44
3. An Incomplete Theory 47
4. Incomplete Differential Calculus 49
5. Inconsistent Differential Calculus 55
6. Integration 56
7. Conclusion 58
vi
SIX INCONSISTENT CONTINUOUS FUNCTIONS
1. Introduction 59
2. Functionality 60
3. Logic 63
I
SEVEN THE DELTA FUNCTION
1. Introduction 67
2. Functionality 68
EIGHT INCONSISTENT SYSTEMS OF LINEAR EQUATIONS
1. Introduction 73
2. The Inconsistent Case 73
3. Control Theory 77
4. Applications, Problems and Special Cases 80
NINE PROJECTIVE SPACES
1. Introduction 83
2. Vector Spaces 84
3. Projective Geometry 86
4. Projective Geometry Modulo Infinity 88
TEN TOPOLOGY 93
ELEVEN CATEGORY THEORY
(with Peter Lavers)
1. Introduction 101
2. Closed Set Logic 103
3. Propositional Logic in a Category 104
4. Implication 109
5. Quantification Theory 112
6. Conclusion 114
TWELVE CLOSED SET SHEAVES AND THEIR CATEGORIES vii
(William James)
1. Introduction 115
2. Pretopologies and Topologies for Categories 116
3. Subobject Classifiers 120
4. Closed Set Sheaves 123
THIRTEEN DUALITY 125
FOURTEEN FOUNDATIONS: PROVABILITY, TRUTH AND SETS
(with Joshua Cole)
1. Introduction 129
2. Provability
2.1 Consistent Preliminaries 130
2.2 The Inconsistent Case 132
3. Truth
3.1 The Fixed Point Method 135
3.2 The Fixed Point Method Applied to Truth Theory 137
3.3 The Proof that Fixed Points Model the T scheme 139
3.4 The Proof that there are Fixed Points 141
4. The Fixed Point Method Applied to Set Theory 141
5. Further Applications 146
BIBLIOGRAPHY 147
INDEX OF DEFINITIONS AND NAMES 152
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| id | DE-604.BV010119467 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:46:51Z |
| institution | BVB |
| isbn | 0792331869 |
| language | English |
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| physical | IX, 155 S. |
| publishDate | 1995 |
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| publisher | Kluwer |
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| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spelling | Mortensen, Chris Verfasser aut Inconsistent mathematics by Chris Mortensen Dordrecht u.a. Kluwer 1995 IX, 155 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 312 The theory of inconsistency has been growing steadily over the last two decades. One focus has been philosophical issues arising from the paradoxes of set theory and semantics. A second focus has been the study of paraconsistent, or inconsistency-tolerant, logics. A third focus has been the application of paraconsistent logics to problems in artificial intelligence. This book focuses on a fourth aspect, the construction of mathematical theories in which contradictions occur, and the investigation of their properties. The inconsistent approach provides a distinctive perspective on the various number systems, order, differential and integral calculus, discontinuous changes, inconsistent systems of linear equations, projective geometry, topology and category theory. The final chapter outlines several known results concerning paradoxes in the foundations of set theory and semantics The book begins with an informal chapter which summarises the main results nontechnically, and draws philosophical implications from them. This volume will be of interest of advanced undergraduates, graduate students and professionals in the areas of logic, philosophy, mathematics and theoretical computer science Logica gtt Logik Inconsistency (Logic) Logic, Symbolic and mathematical Parakonsistente Logik (DE-588)4226190-9 gnd rswk-swf Parakonsistente Logik (DE-588)4226190-9 s DE-604 Mathematics and its applications 312 (DE-604)BV008163334 312 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006719688&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mortensen, Chris Inconsistent mathematics Mathematics and its applications Logica gtt Logik Inconsistency (Logic) Logic, Symbolic and mathematical Parakonsistente Logik (DE-588)4226190-9 gnd |
| subject_GND | (DE-588)4226190-9 |
| title | Inconsistent mathematics |
| title_auth | Inconsistent mathematics |
| title_exact_search | Inconsistent mathematics |
| title_full | Inconsistent mathematics by Chris Mortensen |
| title_fullStr | Inconsistent mathematics by Chris Mortensen |
| title_full_unstemmed | Inconsistent mathematics by Chris Mortensen |
| title_short | Inconsistent mathematics |
| title_sort | inconsistent mathematics |
| topic | Logica gtt Logik Inconsistency (Logic) Logic, Symbolic and mathematical Parakonsistente Logik (DE-588)4226190-9 gnd |
| topic_facet | Logica Logik Inconsistency (Logic) Logic, Symbolic and mathematical Parakonsistente Logik |
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