Realizability: an introduction to its categorical side
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| Format: | Book |
| Language: | English |
| Published: |
Amsterdam
North-Holland
2008
|
| Edition: | 1. ed. |
| Series: | Studies in logic and the foundations of mathematics
152 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 310 S. |
| ISBN: | 9780444515841 |
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| adam_text | REALIZABILITY: AN INTRODUCTION TO ITS CATEGORICAL SIDE JAAP VAN OOSTEN
DEPARTMENT OF MATHEMATICS UTRECHT UNIVERSITY ELSEVIER AMSTERDAM . BOSTON
.HEIDELBERG * LONDON * NEW YORK . OXFORD PARIS * SAN DIEGO . SAN
FRANCISCO * SINGAPORE * SYDNEY . TOKYO CONTENTS PREFACE V INTRODUCTION
IX 1 PARTIAL COMBINATORY ALGEBRAS 1 1.1 BASIC DEFINITIONS 1 1.1.1
PAIRING, BOOLEANS AND DEFINITION BY CASES 5 1.2 ^(^-VALUED PREDICATES 5
1.3 FURTHER PROPERTIES; RECURSION THEORY 11 1.3.1 RECURSION THEORY IN
PEAS 11 1.4 EXAMPLES OF PEAS 15 1.4.1 KLEENE S FIRST MODEL 15 1.4.2
RELATIVIZED RECURSION 15 1.4.3 KLEENE S SECOND MODEL 15 1.4.4 K.2
GENERALIZED 17 1.4.5 SEQUENTIAL COMPUTATIONS 18 1.4.6 THE GRAPH MODEL
V(TO) 20 1.4.7 GRAPH MODELS 21 1.4.8 DOMAIN MODELS 22 1.4.9 RELATIVIZED
MODELS 22 1.4.10 TERM MODELS 23 1.4.11 PITTS CONSTRUCTION 23 1.4.12
MODELS OF ARITHMETIC 23 1.5 MORPHISMS AND ASSEMBLIES 24 1.6 APPLICATIVE
MORPHISMS AND S-FUNCTORS 30 XIN XIV CONTENTS 1.7 DECIDABLE APPLICATIVE
MORPHISMS 35 1.8 ORDER-PEAS 40 2 REALIZABILITY TRIPOSES AND TOPOSES 49
2.1 TRIPOSES 49 2.1.1 PREORDER-ENRICHED CATEGORIES 49 2.1.2 TRIPOSES:
DEFINITION AND BASIC PROPERTIES 51 2.1.3 INTERPRETATION OF LANGUAGES IN
TRIPOSES 55 2.1.4 A FEW USEFUL FACTS 59 2.2 THE TRIPOS-TO-TOPOS
CONSTRUCTION 64 2.3 INTERNAL LOGIC OF C[P] REDUCED TO THE LOGIC OF P 69
2.4 THE CONSTANT OBJECTS FUNCTOR 73 2.5 GEOMETRIC MORPHISMS 82 2.5.1
GEOMETRIC MORPHISMS OF TOPOSES 82 2.5.2 GEOMETRIC MORPHISMS OF TRIPOSES
86 2.5.3 GEOMETRIC MORPHISMS BETWEEN REALIZABILITY TRIPOSES ON SET 92
2.5.4 INCLUSIONS OF TRIPOSES AND TOPOSES 95 2.6 EXAMPLES OF TRIPOSES AND
INCLUSIONS OF TRIPOSES 98 2.6.1 SUBLOCALES 98 2.6.2 ORDER-PEAS 98 2.6.3
SET AS A SUBTOPOS OF RT(A) 99 2.6.4 RELATIVE RECURSION 100 2.6.5
ORDER-PEAS WITH THE PASTING PROPERTY 101 2.6.6 EXTENSIONAL REALIZABILITY
102 2.6.7 MODIFIED REALIZABILITY 102 2.6.8 LIFSCHITZ REALIZABILITY 103
2.6.9 RELATIVE REALIZABILITY 106 2.6.10 DEFINABLE SUBTRIPOSES 107 2.7
ITERATION 109 2.8 GLUEING OF TRIPOSES ILL 3 THE EFFECTIVE TOPOS 115 3.1
RECAPITULATION AND ARITHMETIC IN SFF 115 3.1.1 SECOND-ORDER ARITHMETIC
IN SFF 125 3.1.2 THIRD-ORDER ARITHMETIC IN SFF 131 CONTENTS XV 3.2 SOME
SPECIAL OBJECTS AND ARROWS IN SFF 132 3.2.1 CLOSED AND DENSE SUBOBJECTS
132 3.2.2 INFINITE COPRODUCTS AND PRODUCTS 133 3.2.3 PROJECTIVE AND
INTERNALLY PROJECTIVE OBJECTS, AND CHOICE PRINCIPLES 134 3.2.4 SFF AS A
UNIVERSAL CONSTRUCTION 138 3.2.5 REAL NUMBERS IN SFF 140 3.2.6 DISCRETE
AND MODEST OBJECTS 143 3.2.7 DECIDABLE AND SEMIDECIDABLE SUBOBJECTS 148
3.3 SOME ANALYSIS IN SFF 154 3.3.1 GENERAL FACTS ABOUT R 155 3.3.2
SPECKER SEQUENCES AND SINGULAR COVERINGS 157 3.3.3 REAL-VALUED FUNCTIONS
159 3.4 DISCRETE FAMILIES AND UNIFORM MAPS 162 3.4.1 WEAKLY COMPLETE
INTERNAL CATEGORIES IN SFF . . . .178 3.5 SET THEORY IN SFF 193 3.5.1
THE MCCARTY MODEL FOR IZF 193 3.5.2 THE LUBARSKY-STREICHER-VAN DEN BERG
MODEL FOR CZF . 211 3.5.3 WELL-FOUNDED TREES AND TV-TYPES IN SFF 212 3.6
SYNTHETIC DOMAIN THEORY IN SFF 214 3.6.1 COMPLETE PARTIAL ORDERS 215
3.6.2 THE SYNTHETIC APPROACH 218 3.6.3 ELEMENTS OF SYNTHETIC DOMAIN
THEORY 219 3.6.4 MODELS FOR SDT IN SFF 228 3.7 SYNTHETIC COMPUTABILITY
THEORY IN SFF 230 3.8 GENERAL COMMENTS ABOUT THE EFFECTIVE TOPOS 234
3.8.1 ANALOGY BETWEEN V AND THE YONEDA EMBEDDING . . 235 3.8.2 SMALL
DENSE SUBCATEGORIES IN SFF 239 3.8.3 IDEMPOTENCE OF REALIZABILITY 245 4
VARIATIONS 255 4.1 EXTENSIONAL REALIZABILITY 255 4.1.1 EXT AS EXACT
COMPLETION? 262 4.2 MODIFIED REALIZABILITY 263 XVI CONTENTS 4.3 FUNCTION
REALIZABILITY 268 4.4 LIFSCHITZ REALIZABILITY 274 4.5 RELATIVE
REALIZABILITY 277 4.6 REALIZABILITY TOPOSES OVER OTHER TOPOSES 283 4.6.1
THE FREE TOPOS WITH NNO 283 4.6.2 A SHEAF MODEL OF REALIZABILITY 287
BIBLIOGRAPHY 291 INDEX 305
|
| adam_txt |
REALIZABILITY: AN INTRODUCTION TO ITS CATEGORICAL SIDE JAAP VAN OOSTEN
DEPARTMENT OF MATHEMATICS UTRECHT UNIVERSITY ELSEVIER AMSTERDAM . BOSTON
.HEIDELBERG * LONDON * NEW YORK . OXFORD PARIS * SAN DIEGO .' SAN
FRANCISCO * SINGAPORE * SYDNEY . TOKYO CONTENTS PREFACE V INTRODUCTION
IX 1 PARTIAL COMBINATORY ALGEBRAS 1 1.1 BASIC DEFINITIONS 1 1.1.1
PAIRING, BOOLEANS AND DEFINITION BY CASES 5 1.2 ^(^-VALUED PREDICATES 5
1.3 FURTHER PROPERTIES; RECURSION THEORY 11 1.3.1 RECURSION THEORY IN
PEAS 11 1.4 EXAMPLES OF PEAS 15 1.4.1 KLEENE'S FIRST MODEL 15 1.4.2
RELATIVIZED RECURSION 15 1.4.3 KLEENE'S SECOND MODEL 15 1.4.4 K.2
GENERALIZED 17 1.4.5 SEQUENTIAL COMPUTATIONS 18 1.4.6 THE GRAPH MODEL
V(TO) 20 1.4.7 GRAPH MODELS 21 1.4.8 DOMAIN MODELS 22 1.4.9 RELATIVIZED
MODELS 22 1.4.10 TERM MODELS 23 1.4.11 PITTS' CONSTRUCTION 23 1.4.12
MODELS OF ARITHMETIC 23 1.5 MORPHISMS AND ASSEMBLIES 24 1.6 APPLICATIVE
MORPHISMS AND S-FUNCTORS 30 XIN XIV CONTENTS 1.7 DECIDABLE APPLICATIVE
MORPHISMS 35 1.8 ORDER-PEAS 40 2 REALIZABILITY TRIPOSES AND TOPOSES 49
2.1 TRIPOSES 49 2.1.1 PREORDER-ENRICHED CATEGORIES 49 2.1.2 TRIPOSES:
DEFINITION AND BASIC PROPERTIES 51 2.1.3 INTERPRETATION OF LANGUAGES IN
TRIPOSES 55 2.1.4 A FEW USEFUL FACTS 59 2.2 THE TRIPOS-TO-TOPOS
CONSTRUCTION 64 2.3 INTERNAL LOGIC OF C[P] REDUCED TO THE LOGIC OF P 69
2.4 THE 'CONSTANT OBJECTS' FUNCTOR 73 2.5 GEOMETRIC MORPHISMS 82 2.5.1
GEOMETRIC MORPHISMS OF TOPOSES 82 2.5.2 GEOMETRIC MORPHISMS OF TRIPOSES
86 2.5.3 GEOMETRIC MORPHISMS BETWEEN REALIZABILITY TRIPOSES ON SET 92
2.5.4 INCLUSIONS OF TRIPOSES AND TOPOSES 95 2.6 EXAMPLES OF TRIPOSES AND
INCLUSIONS OF TRIPOSES 98 2.6.1 SUBLOCALES 98 2.6.2 ORDER-PEAS 98 2.6.3
SET AS A SUBTOPOS OF RT(A) 99 2.6.4 RELATIVE RECURSION 100 2.6.5
ORDER-PEAS WITH THE PASTING PROPERTY 101 2.6.6 EXTENSIONAL REALIZABILITY
102 2.6.7 MODIFIED REALIZABILITY 102 2.6.8 LIFSCHITZ REALIZABILITY 103
2.6.9 RELATIVE REALIZABILITY 106 2.6.10 DEFINABLE SUBTRIPOSES 107 2.7
ITERATION 109 2.8 GLUEING OF TRIPOSES ILL 3 THE EFFECTIVE TOPOS 115 3.1
RECAPITULATION AND ARITHMETIC IN SFF 115 3.1.1 SECOND-ORDER ARITHMETIC
IN SFF 125 3.1.2 THIRD-ORDER ARITHMETIC IN SFF 131 CONTENTS XV 3.2 SOME
SPECIAL OBJECTS AND ARROWS IN SFF 132 3.2.1 CLOSED AND DENSE SUBOBJECTS
132 3.2.2 INFINITE COPRODUCTS AND PRODUCTS 133 3.2.3 PROJECTIVE AND
INTERNALLY PROJECTIVE OBJECTS, AND CHOICE PRINCIPLES 134 3.2.4 SFF AS A
UNIVERSAL CONSTRUCTION 138 3.2.5 REAL NUMBERS IN SFF 140 3.2.6 DISCRETE
AND MODEST OBJECTS 143 3.2.7 DECIDABLE AND SEMIDECIDABLE SUBOBJECTS 148
3.3 SOME ANALYSIS IN SFF 154 3.3.1 GENERAL FACTS ABOUT R 155 3.3.2
SPECKER SEQUENCES AND SINGULAR COVERINGS 157 3.3.3 REAL-VALUED FUNCTIONS
159 3.4 DISCRETE FAMILIES AND UNIFORM MAPS 162 3.4.1 WEAKLY COMPLETE
INTERNAL CATEGORIES IN SFF . . . .178 3.5 SET THEORY IN SFF 193 3.5.1
THE MCCARTY MODEL FOR IZF 193 3.5.2 THE LUBARSKY-STREICHER-VAN DEN BERG
MODEL FOR CZF . 211 3.5.3 WELL-FOUNDED TREES AND TV-TYPES IN SFF 212 3.6
SYNTHETIC DOMAIN THEORY IN SFF 214 3.6.1 COMPLETE PARTIAL ORDERS 215
3.6.2 THE SYNTHETIC APPROACH 218 3.6.3 ELEMENTS OF SYNTHETIC DOMAIN
THEORY 219 3.6.4 MODELS FOR SDT IN SFF 228 3.7 SYNTHETIC COMPUTABILITY
THEORY IN SFF 230 3.8 GENERAL COMMENTS ABOUT THE EFFECTIVE TOPOS 234
3.8.1 ANALOGY BETWEEN V AND THE YONEDA EMBEDDING . . 235 3.8.2 SMALL
DENSE SUBCATEGORIES IN SFF 239 3.8.3 IDEMPOTENCE OF REALIZABILITY 245 4
VARIATIONS " 255 4.1 EXTENSIONAL REALIZABILITY 255 4.1.1 EXT AS EXACT
COMPLETION? 262 4.2 MODIFIED REALIZABILITY 263 XVI CONTENTS 4.3 FUNCTION
REALIZABILITY 268 4.4 LIFSCHITZ REALIZABILITY 274 4.5 RELATIVE
REALIZABILITY 277 4.6 REALIZABILITY TOPOSES OVER OTHER TOPOSES 283 4.6.1
THE FREE TOPOS WITH NNO 283 4.6.2 A SHEAF MODEL OF REALIZABILITY 287
BIBLIOGRAPHY 291 INDEX 305 |
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| author_facet | Oosten, Jaap van |
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| index_date | 2024-07-02T22:16:58Z |
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| institution | BVB |
| isbn | 9780444515841 |
| language | English |
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| series2 | Studies in logic and the foundations of mathematics |
| spelling | Oosten, Jaap van Verfasser aut Realizability an introduction to its categorical side Jaap van Oosten 1. ed. Amsterdam North-Holland 2008 XVI, 310 S. txt rdacontent n rdamedia nc rdacarrier Studies in logic and the foundations of mathematics 152 Realisierung Mathematik (DE-588)4758721-0 gnd rswk-swf Realisierung Mathematik (DE-588)4758721-0 s DE-604 Studies in logic and the foundations of mathematics 152 (DE-604)BV000893472 152 HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016777359&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Oosten, Jaap van Realizability an introduction to its categorical side Studies in logic and the foundations of mathematics Realisierung Mathematik (DE-588)4758721-0 gnd |
| subject_GND | (DE-588)4758721-0 |
| title | Realizability an introduction to its categorical side |
| title_auth | Realizability an introduction to its categorical side |
| title_exact_search | Realizability an introduction to its categorical side |
| title_exact_search_txtP | Realizability an introduction to its categorical side |
| title_full | Realizability an introduction to its categorical side Jaap van Oosten |
| title_fullStr | Realizability an introduction to its categorical side Jaap van Oosten |
| title_full_unstemmed | Realizability an introduction to its categorical side Jaap van Oosten |
| title_short | Realizability |
| title_sort | realizability an introduction to its categorical side |
| title_sub | an introduction to its categorical side |
| topic | Realisierung Mathematik (DE-588)4758721-0 gnd |
| topic_facet | Realisierung Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016777359&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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