Computability and logic:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2007
|
| Edition: | 5. ed. |
| Subjects: | |
| Online Access: | Table of contents only Publisher description Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke Includes bibliographical references (p. 341) and index |
| Physical Description: | XIII, 350 S. Ill. |
| ISBN: | 9780521877527 0521877520 9780521701464 0521701465 |
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Record in the Search Index
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| adam_text | Titel: Computability and logic
Autor: Boolos, George
Jahr: 2007
Contents
Preface to the Fifth Edition page xi
COMPUTABILITY THEORY
1 Enumerability 3
1.1 Enumerability 3
1.2 Enumerable Sets 7
2 Diagonalization 16
3 Turing Computability 23
4 Uncomputability 35
4.1 The Halting Problem 35
4.2 The Productivity Function 40
5 Abacus Computability 45
5.1 Abacus Machines 45
5.2 Simulating Abacus Machines by Turing Machines 51
5.3 The Scope of Abacus Computability 57
6 Recursive Functions 63
6.1 Primitive Recursive Functions 63
6.2 Minimization 70
7 Recursive Sets and Relations 73
7.1 Recursive Relations 73
7.2 Semirecursive Relations 80
7.3 Further Examples 83
8 Equivalent Definitions of Computability 88
8.1 Coding Turing Computations 88
8.2 Universal Turing Machines 94
8.3 Recursively Enumerable Sets 96
vj¡¡ CONTENTS
BASIC METALOGIC
9 A Précis of First-Order Logic: Syntax 101
9.1 First-Order Logic 101
9.2 Syntax 106
10 A Précis of First-Order Logic: Semantics 114
10.1 Semantics 114
1 IQ
10.2 Metalogical Notions liy
11 The Undecidability of First-Order Logic 126
11.1 Logic and Turing Machines 126
11.2 Logic and Primitive Recursive Functions 1 32
12 Models 137
12.1 The Size and Number of Models l^7
12.2 Equivalence Relations 1^2
12.3 The Löwenheim-Skolem and Compactness Theorems 146
13 The Existence of Models 153
13.1 Outline of the Proof I53
13.2 The First Stage of the Proof I56
13.3 The Second Stage of the Proof I57
13.4 The Third Stage of the Proof 160
13.5 Nonenumerable Languages 162
14 Proofs and Completeness 166
14.1 Sequent Calculus 166
14.2 Soundness and Completeness 174
14.3 Other Proof Procedures and Hubert s Thesis 179
15 Arithmetization 187
15.1 Arithmetization of Syntax 187
15.2 Gödel Numbers 192
15.3 More Gödel Numbers 196
16 Representability of Recursive Functions 199
16.1 Arithmetical Definability 199
16.2 Minimal Arithmetic and Representability 207
16.3 Mathematical Induction 212
16.4 Robinson Arithmetic 216
17 Indefinability, Undecidability, Incompleteness 220
17.1 The Diagonal Lemma and the Limitative Theorems 220
17.2 Undecidable Sentences 224
17.3 Undecidable Sentences without the Diagonal Lemma 226
18 The Unprovability of Consistency 232
CONTENTS ix
FURTHER TOPICS
19 Normal Forms 243
19.1 Disjunctive and Prenex Normal Forms 243
19.2 Skolem Normal Form 247
19.3 Herbrand s Theorem 253
19.4 Eliminating Function Symbols and Identity 255
20 The Craig Interpolation Theorem 260
20.1 Craig s Theorem and Its Proof 260
20.2 Robinson s Joint Consistency Theorem 264
20.3 Beth s Definability Theorem 265
21 Monadic and Dyadic Logic 270
21.1 Solvable and Unsolvable Decision Problems 270
21.2 Monadic Logic 273
21.3 Dyadic Logic 275
22 Second-Order Logic 279
23 Arithmetical Definability 286
23.1 Arithmetical Definability and Truth 286
23.2 Arithmetical Definability and Forcing 289
24 Decidability of Arithmetic without Multiplication 295
25 Nonstandard Models 302
25.1 Order in Nonstandard Models 302
25.2 Operations in Nonstandard Models 306
25.3 Nonstandard Models of Analysis 312
26 Ramsey s Theorem 319
26.1 Ramsey s Theorem: Finitary and Infinitary 319
26.2 Konig s Lemma 322
27 Modal Logic and Provability 327
27.1 Modal Logic 327
27.2 The Logic of Provability 334
27.3 The Fixed Point and Normal Form Theorems 337
Annotated Bibliography 341
Index 343
|
| adam_txt |
Titel: Computability and logic
Autor: Boolos, George
Jahr: 2007
Contents
Preface to the Fifth Edition page xi
COMPUTABILITY THEORY
1 Enumerability 3
1.1 Enumerability 3
1.2 Enumerable Sets 7
2 Diagonalization 16
3 Turing Computability 23
4 Uncomputability 35
4.1 The Halting Problem 35
4.2 The Productivity Function 40
5 Abacus Computability 45
5.1 Abacus Machines 45
5.2 Simulating Abacus Machines by Turing Machines 51
5.3 The Scope of Abacus Computability 57
6 Recursive Functions 63
6.1 Primitive Recursive Functions 63
6.2 Minimization 70
7 Recursive Sets and Relations 73
7.1 Recursive Relations 73
7.2 Semirecursive Relations 80
7.3 Further Examples 83
8 Equivalent Definitions of Computability 88
8.1 Coding Turing Computations 88
8.2 Universal Turing Machines 94
8.3 Recursively Enumerable Sets 96
vj¡¡ CONTENTS
BASIC METALOGIC
9 A Précis of First-Order Logic: Syntax 101
9.1 First-Order Logic 101
9.2 Syntax 106
10 A Précis of First-Order Logic: Semantics 114
10.1 Semantics 114
1 IQ
10.2 Metalogical Notions liy
11 The Undecidability of First-Order Logic 126
11.1 Logic and Turing Machines 126
11.2 Logic and Primitive Recursive Functions 1 32
12 Models 137
12.1 The Size and Number of Models l^7
12.2 Equivalence Relations 1^2
12.3 The Löwenheim-Skolem and Compactness Theorems 146
13 The Existence of Models 153
13.1 Outline of the Proof I53
13.2 The First Stage of the Proof I56
13.3 The Second Stage of the Proof I57
13.4 The Third Stage of the Proof 160
13.5 Nonenumerable Languages 162
14 Proofs and Completeness 166
14.1 Sequent Calculus 166
14.2 Soundness and Completeness 174
14.3 Other Proof Procedures and Hubert's Thesis 179
15 Arithmetization 187
15.1 Arithmetization of Syntax 187
15.2 Gödel Numbers 192
15.3 More Gödel Numbers 196
16 Representability of Recursive Functions 199
16.1 Arithmetical Definability 199
16.2 Minimal Arithmetic and Representability 207
16.3 Mathematical Induction 212
16.4 Robinson Arithmetic 216
17 Indefinability, Undecidability, Incompleteness 220
17.1 The Diagonal Lemma and the Limitative Theorems 220
17.2 Undecidable Sentences 224
17.3 Undecidable Sentences without the Diagonal Lemma 226
18 The Unprovability of Consistency 232
CONTENTS ix
FURTHER TOPICS
19 Normal Forms 243
19.1 Disjunctive and Prenex Normal Forms 243
19.2 Skolem Normal Form 247
19.3 Herbrand's Theorem 253
19.4 Eliminating Function Symbols and Identity 255
20 The Craig Interpolation Theorem 260
20.1 Craig's Theorem and Its Proof 260
20.2 Robinson's Joint Consistency Theorem 264
20.3 Beth's Definability Theorem 265
21 Monadic and Dyadic Logic 270
21.1 Solvable and Unsolvable Decision Problems 270
21.2 Monadic Logic 273
21.3 Dyadic Logic 275
22 Second-Order Logic 279
23 Arithmetical Definability 286
23.1 Arithmetical Definability and Truth 286
23.2 Arithmetical Definability and Forcing 289
24 Decidability of Arithmetic without Multiplication 295
25 Nonstandard Models 302
25.1 Order in Nonstandard Models 302
25.2 Operations in Nonstandard Models 306
25.3 Nonstandard Models of Analysis 312
26 Ramsey's Theorem 319
26.1 Ramsey's Theorem: Finitary and Infinitary 319
26.2 Konig's Lemma 322
27 Modal Logic and Provability 327
27.1 Modal Logic 327
27.2 The Logic of Provability 334
27.3 The Fixed Point and Normal Form Theorems 337
Annotated Bibliography 341
Index 343 |
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| spelling | Boolos, George 1940-1996 Verfasser (DE-588)171984196 aut Computability and logic George S. Boolos ; John P. Burgess ; Richard C. Jeffrey 5. ed. Cambridge [u.a.] Cambridge Univ. Press 2007 XIII, 350 S. Ill. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Includes bibliographical references (p. 341) and index Computable functions Recursive functions Logic, Symbolic and mathematical Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Berechenbarkeit (DE-588)4138368-0 gnd rswk-swf Berechenbarkeit (DE-588)4138368-0 s Mathematische Logik (DE-588)4037951-6 s DE-604 Burgess, John P. 1948- Verfasser (DE-588)139005250 aut Jeffrey, Richard C. 1926-2002 Verfasser (DE-588)119344491 aut http://www.loc.gov/catdir/toc/ecip0715/2007014225.html Table of contents only http://www.loc.gov/catdir/enhancements/fy0729/2007014225-d.html Publisher description HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016602356&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Boolos, George 1940-1996 Burgess, John P. 1948- Jeffrey, Richard C. 1926-2002 Computability and logic Computable functions Recursive functions Logic, Symbolic and mathematical Mathematische Logik (DE-588)4037951-6 gnd Berechenbarkeit (DE-588)4138368-0 gnd |
| subject_GND | (DE-588)4037951-6 (DE-588)4138368-0 |
| title | Computability and logic |
| title_auth | Computability and logic |
| title_exact_search | Computability and logic |
| title_exact_search_txtP | Computability and logic |
| title_full | Computability and logic George S. Boolos ; John P. Burgess ; Richard C. Jeffrey |
| title_fullStr | Computability and logic George S. Boolos ; John P. Burgess ; Richard C. Jeffrey |
| title_full_unstemmed | Computability and logic George S. Boolos ; John P. Burgess ; Richard C. Jeffrey |
| title_short | Computability and logic |
| title_sort | computability and logic |
| topic | Computable functions Recursive functions Logic, Symbolic and mathematical Mathematische Logik (DE-588)4037951-6 gnd Berechenbarkeit (DE-588)4138368-0 gnd |
| topic_facet | Computable functions Recursive functions Logic, Symbolic and mathematical Mathematische Logik Berechenbarkeit |
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