The structure of models of Peano arithmetic:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford
Clarendon Press
2006
|
| Series: | Oxford logic guides
50 |
| Subjects: | |
| Online Access: | Table of contents only Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (S. [295]-306) and indexes |
| Physical Description: | XIV, 311 S. 24 cm |
| ISBN: | 9780198568278 0198568274 |
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| 245 | 1 | 0 | |a The structure of models of Peano arithmetic |c Roman Kossak ; James H. Schmerl |
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| 300 | |a XIV, 311 S. |c 24 cm | ||
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Record in the Search Index
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| adam_text | Titel: The structure of models of Peano arithmetic
Autor: Kossak, Roman
Jahr: 2006
CONTENTS
1 Basics 1
1.1 Notation and basic definitions 1
1.2 Skolem closures 4
1.3 End extensions and cofinal extensions 5
1.4 Coding bounded sets and classes 6
1.5 Standard systems 8
1.6 Types 9
1.7 Blass-Gaifman and Ehrenfeucht lemmas 10
1.8 Recursive saturation and arithmetic saturation 12
1.9 Satisfaction classes and resplendency 14
1.10 Cuts and gaps in recursively saturated models 16
1.11 Truth definitions and restricted saturation 18
1.12 Arithmetized Completeness Theorem 19
1.13 Friedman s Embedding Theorem 20
1.14 Exercises 21
1.15 Remarks References 22
2 Extensions 25
2.1 Simple extensions 25
2.1.1 Minimal extensions 27
2.1.2 Superminimal extensions 29
2.1.3 Greatest common initial segments 31
2.2 The MacDowell-Specker Theorem 33
2.2.1 Superminimal conservative extensions 37
2.2.2 Rather classless models 37
2.2.3 Ramsey s Theorem in ACAo 38
2.3 Amalgamations 39
2.4 Nonelementary extensions 43
2.5 Exercises 45
2.6 Remarks References 47
3 Minimal and Other Types 49
3.1 Types related to indiscernibility 49
3.1.1 Indiscernible types 49
3.1.2 n-indiscernible types 53
3.1.3 End-extensional types 55
3.1.4 Rare types 56
3.2 Minimal types 59
3.2.1 Selective types 59
61
65
66
67
69
72
73
78
84
88
89
89
94
97
102
105
122
127
130
132
135
135
136
137
139
142
144
145
149
151
156
157
158
158
160
160
161
162
163
165
CONTENTS
3.2.2 Characterizing minimal types
3.2.3 An example
3.3 Canonical extensions
3.3.1 Products of types
3.3.2 The automorphism group
3.3.3 The substructure lattice
3.4 Resolute types
3.5 The Paris-Mills theorems
3.6 Exercises
3.7 Remarks References
Substructure Lattices
4.1 Lattices
4.2 Substructure lattices
4.3 Finite distributive lattices, I
4.4 Finite distributive lattices, II
4.5 Finite lattices
4.6 The pentagon lattice
4.7 Infinite distributive lattices
4.8 Exercises
4.0 Remarks Sz References
How to Control Types
5.1 Solid bases and AH-sets
5.1.1 Controlling indiscernibles and
automorphisms
5.1.2 AH-sets
5.1.3 The proof
5.1.4 True arithmetic
5.2 Omitting indiscernibles
5.3 Ilanf numbers
5.4 The automorphism group
5.5 Indiscernible generators
5.6 Exercises
5.7 Remarks ; References
Generics and Forcing
0.1 Generics
6.2 Forcing
6.2.1 Definition
6.2.2 n-Generics
6.2.3 Prime expansions
6.2.4 The Low Basis Theorem
6.3 Product forcing
CONTENTS xiii
6.4 MacDowell-Specker vs the uncountable 167
6.4.1 No end extension 168
6.4.2 Extensions with mutual generics 169
6.4.3 Getting many classes 172
6.5 Perfect generics 175
6.6 Exercises 178
6.7 Remarks References 179
7 Cuts 180
7.1 Semiregular cuts 181
7.1.1 Semiregularity and WKLo 183
7.2 Regular cuts 184
7.3 Many faces of strongness 189
7.4 Why PA? 194
7.4.1 Schemes axiomatizing arithmetic 196
7.5 Exercises 200
7.6 Remarks References 201
8 Automorphisms of Recursively Saturated Models 202
8.1 Moving undefinable elements 202
8.2 Moving cuts and classes 203
8.3 Moving gaps 205
8.4 Back-and-forth 206
8.5 Extending automorphisms 210
8.6 Maximal automorphisms 214
8.7 Fixing strong cuts 217
8.8 Topology on the automorphism group 219
8.9 Maximal point stabilizers 221
8.10 Arithmetic saturation and open subgroups 223
8.11 Exercises 225
8.12 Remarks References 227
9 Automorphism Groups of Recursively
Saturated Models 230
9.1 Generic automorphisms 230
9.2 Dense conjugacy classes 235
9.3 Small index property 239
9.3.1 The cofinality of the automorphism group 242
9.3.2 Property FA 244
9.4 Coding the standard system 245
9.5 The spectrum of automorphism groups 251
9.6 Exercises 253
9.7 Remarks References 254
xiv
CONTENTS
10 oq-like Models 256
10.1 cJi-Like recursively saturated models 256
10.2 Similar nonisomorphic models 260
10.3 Finitely determinate structures and PA(aa) 265
10.4 Ramsey quantifiers and PA(Q2) 269
10.5 Rigid recursively saturated models 273
10.6 Isomorphic + nonisomorphic x 275
10.7 Exercises 276
10.8 Remarks References 278
11 Order Types 281
11.1 On (k, re)-cuts 281
11.2 Saturation of the order rcduct 282
11.2.1 Canonical codes 283
11.2.2 Ki-saturation 283
11.2.3 ^-saturation for k Hi 284
11.3 Exercises 287
11.4 Remarks References 288
12 Twenty Questions 289
References 295
Index of Names 307
Index of Terms 399
|
| adam_txt |
Titel: The structure of models of Peano arithmetic
Autor: Kossak, Roman
Jahr: 2006
CONTENTS
1 Basics 1
1.1 Notation and basic definitions 1
1.2 Skolem closures 4
1.3 End extensions and cofinal extensions 5
1.4 Coding bounded sets and classes 6
1.5 Standard systems 8
1.6 Types 9
1.7 Blass-Gaifman and Ehrenfeucht lemmas 10
1.8 Recursive saturation and arithmetic saturation 12
1.9 Satisfaction classes and resplendency 14
1.10 Cuts and gaps in recursively saturated models 16
1.11 Truth definitions and restricted saturation 18
1.12 Arithmetized Completeness Theorem 19
1.13 Friedman's Embedding Theorem 20
1.14 Exercises 21
1.15 Remarks References 22
2 Extensions 25
2.1 Simple extensions 25
2.1.1 Minimal extensions 27
2.1.2 Superminimal extensions 29
2.1.3 Greatest common initial segments 31
2.2 The MacDowell-Specker Theorem 33
2.2.1 Superminimal conservative extensions 37
2.2.2 Rather classless models 37
2.2.3 Ramsey's Theorem in ACAo 38
2.3 Amalgamations 39
2.4 Nonelementary extensions 43
2.5 Exercises 45
2.6 Remarks References 47
3 Minimal and Other Types 49
3.1 Types related to indiscernibility 49
3.1.1 Indiscernible types 49
3.1.2 n-indiscernible types 53
3.1.3 End-extensional types 55
3.1.4 Rare types 56
3.2 Minimal types 59
3.2.1 Selective types 59
61
65
66
67
69
72
73
78
84
88
89
89
94
97
102
105
122
127
130
132
135
135
136
137
139
142
144
145
149
151
156
157
158
158
160
160
161
162
163
165
CONTENTS
3.2.2 Characterizing minimal types
3.2.3 An example
3.3 Canonical extensions
3.3.1 Products of types
3.3.2 The automorphism group
3.3.3 The substructure lattice
3.4 Resolute types
3.5 The Paris-Mills theorems
3.6 Exercises
3.7 Remarks References
Substructure Lattices
4.1 Lattices
4.2 Substructure lattices
4.3 Finite distributive lattices, I
4.4 Finite distributive lattices, II
4.5 Finite lattices
4.6 The pentagon lattice
4.7 Infinite distributive lattices
4.8 Exercises
4.0 Remarks Sz References
How to Control Types
5.1 Solid bases and AH-sets
5.1.1 Controlling indiscernibles and
automorphisms
5.1.2 AH-sets
5.1.3 The proof
5.1.4 True arithmetic
5.2 Omitting indiscernibles
5.3 Ilanf numbers
5.4 The automorphism group
5.5 Indiscernible generators
5.6 Exercises
5.7 Remarks ; References
Generics and Forcing
0.1 Generics
6.2 Forcing
6.2.1 Definition
6.2.2 n-Generics
6.2.3 Prime expansions
6.2.4 The Low Basis Theorem
6.3 Product forcing
CONTENTS xiii
6.4 MacDowell-Specker vs the uncountable 167
6.4.1 No end extension 168
6.4.2 Extensions with mutual generics 169
6.4.3 Getting many classes 172
6.5 Perfect generics 175
6.6 Exercises 178
6.7 Remarks References 179
7 Cuts 180
7.1 Semiregular cuts 181
7.1.1 Semiregularity and WKLo 183
7.2 Regular cuts 184
7.3 Many faces of strongness 189
7.4 Why PA? 194
7.4.1 Schemes axiomatizing arithmetic 196
7.5 Exercises 200
7.6 Remarks References 201
8 Automorphisms of Recursively Saturated Models 202
8.1 Moving undefinable elements 202
8.2 Moving cuts and classes 203
8.3 Moving gaps 205
8.4 Back-and-forth 206
8.5 Extending automorphisms 210
8.6 Maximal automorphisms 214
8.7 Fixing strong cuts 217
8.8 Topology on the automorphism group 219
8.9 Maximal point stabilizers 221
8.10 Arithmetic saturation and open subgroups 223
8.11 Exercises 225
8.12 Remarks References 227
9 Automorphism Groups of Recursively
Saturated Models 230
9.1 Generic automorphisms 230
9.2 Dense conjugacy classes 235
9.3 Small index property 239
9.3.1 The cofinality of the automorphism group 242
9.3.2 Property FA 244
9.4 Coding the standard system 245
9.5 The spectrum of automorphism groups 251
9.6 Exercises 253
9.7 Remarks References 254
xiv
CONTENTS
10 oq-like Models 256
10.1 cJi-Like recursively saturated models 256
10.2 Similar nonisomorphic models 260
10.3 Finitely determinate structures and PA(aa) 265
10.4 Ramsey quantifiers and PA(Q2) 269
10.5 Rigid recursively saturated models 273
10.6 Isomorphic + nonisomorphic x 275
10.7 Exercises 276
10.8 Remarks References 278
11 Order Types 281
11.1 On (k, re)-cuts 281
11.2 Saturation of the order rcduct 282
11.2.1 Canonical codes 283
11.2.2 Ki-saturation 283
11.2.3 ^-saturation for k Hi 284
11.3 Exercises 287
11.4 Remarks References 288
12 Twenty Questions 289
References 295
Index of Names 307
Index of Terms 399 |
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| spelling | Kossak, Roman 1953- Verfasser (DE-588)113727199X aut The structure of models of Peano arithmetic Roman Kossak ; James H. Schmerl Oxford Clarendon Press 2006 XIV, 311 S. 24 cm txt rdacontent n rdamedia nc rdacarrier Oxford logic guides 50 Includes bibliographical references (S. [295]-306) and indexes Peano, Giuseppe 1858-1932 Logica gtt Structurele vergelijkingen gtt Logik Logic, Symbolic and mathematical Logic, Symbolic and mathematical Problems, exercises, etc Isomorphisms (Mathematics) Isomorphisms (Mathematics) Problems, exercises, etc Nonstandard-Logik (DE-588)4171988-8 gnd rswk-swf Peano-Arithmetik (DE-588)4290970-3 gnd rswk-swf Nonstandard-Logik (DE-588)4171988-8 s Peano-Arithmetik (DE-588)4290970-3 s DE-604 Schmerl, James Henry Verfasser aut Oxford logic guides 50 (DE-604)BV000013997 50 http://www.loc.gov/catdir/toc/fy0702/2006298586.html Table of contents only HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015053304&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kossak, Roman 1953- Schmerl, James Henry The structure of models of Peano arithmetic Oxford logic guides Peano, Giuseppe 1858-1932 Logica gtt Structurele vergelijkingen gtt Logik Logic, Symbolic and mathematical Logic, Symbolic and mathematical Problems, exercises, etc Isomorphisms (Mathematics) Isomorphisms (Mathematics) Problems, exercises, etc Nonstandard-Logik (DE-588)4171988-8 gnd Peano-Arithmetik (DE-588)4290970-3 gnd |
| subject_GND | (DE-588)4171988-8 (DE-588)4290970-3 |
| title | The structure of models of Peano arithmetic |
| title_auth | The structure of models of Peano arithmetic |
| title_exact_search | The structure of models of Peano arithmetic |
| title_exact_search_txtP | The structure of models of Peano arithmetic |
| title_full | The structure of models of Peano arithmetic Roman Kossak ; James H. Schmerl |
| title_fullStr | The structure of models of Peano arithmetic Roman Kossak ; James H. Schmerl |
| title_full_unstemmed | The structure of models of Peano arithmetic Roman Kossak ; James H. Schmerl |
| title_short | The structure of models of Peano arithmetic |
| title_sort | the structure of models of peano arithmetic |
| topic | Peano, Giuseppe 1858-1932 Logica gtt Structurele vergelijkingen gtt Logik Logic, Symbolic and mathematical Logic, Symbolic and mathematical Problems, exercises, etc Isomorphisms (Mathematics) Isomorphisms (Mathematics) Problems, exercises, etc Nonstandard-Logik (DE-588)4171988-8 gnd Peano-Arithmetik (DE-588)4290970-3 gnd |
| topic_facet | Peano, Giuseppe 1858-1932 Logica Structurele vergelijkingen Logik Logic, Symbolic and mathematical Logic, Symbolic and mathematical Problems, exercises, etc Isomorphisms (Mathematics) Isomorphisms (Mathematics) Problems, exercises, etc Nonstandard-Logik Peano-Arithmetik |
| url | http://www.loc.gov/catdir/toc/fy0702/2006298586.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015053304&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000013997 |
| work_keys_str_mv | AT kossakroman thestructureofmodelsofpeanoarithmetic AT schmerljameshenry thestructureofmodelsofpeanoarithmetic |