Metamathematics of first-order arithmetic:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin u.a.
Springer
1993
|
| Series: | Perspectives in mathematical logic
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 410 - 453 |
| Physical Description: | XIV, 460 S. |
| ISBN: | 3540506322 0387506322 |
Staff View
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| 245 | 1 | 0 | |a Metamathematics of first-order arithmetic |c Petr Hájek ; Pavel Pudlák |
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Record in the Search Index
| _version_ | 1804122393522733056 |
|---|---|
| adam_text | U& BFT,-TFTCT PETR HAEJEK PAVEL PUDLAEK METAMATHEMATICS OF FIRST-ORDER
ARITHMETIC SPRINGER-VERLAG BERLIN HEIDELBERG NEW YORK LONDON PARIS TOKYO
HONG KONG BARCELONA BUDAPEST TABLE OF CONTENTS INTRODUCTION 1
PRELIMINARIES 5 (A) SOME LOGIC 5 (B) THE LANGUAGE OF ARITHMETIC, THE
STANDARD MODEL 12 (C) BEGINNING ARITHMETIZATION OF METAMATHEMATICS 20
PART A CHAPTER I ARITHMETIC AS NUMBER THEORY, SET THEORY AND LOGIC 27
INTRODUCTION 27 1. BASIC DEVELOPMENTS; PARTIAL TRUTH DEFINITIONS 28 (A)
PROPERTIES OF ADDITION AND MULTIPLICATION, DIVISIBILITY AND PRIMES 28
(B) CODING FINITE SETS AND SEQUENCES; THE THEORY ISO(EXP) . . 37 (C)
PROVABLY RECURSIVE FUNCTIONS; THE THEORY IS 44 (D) ARITHMETIZATION OF
METAMATHEMATICS: PARTIAL TRUTH DEFINITIONS 50 2. FRAGMENTS OF
FIRST-ORDER ARITHMETIC 61 (A) INDUCTION AND COLLECTION 61 (B) FURTHER
PRINCIPLES AND FACTS ABOUT FRAGMENTS 67 (C) FINITE AXIOMATIZABILITY;
PARTIAL TRUTH DEFINITIONS FOR RELATIVIZED ARITHMETICAL FORMULAS 77 (D)
RELATIVIZED HIERARCHY IN FRAGMENTS 81 (E) AXIOMATIC SYSTEMS OF
ARITHMETIC WITH NO FUNCTION SYMBOLS . 86 3. FRAGMENTS AND RECURSION
THEORY 89 (A) LIMIT THEOREM 89 (B) LOW BASIS THEOREM 91 (C) INFINITE A
SUBSETS 95 (D) MATIYASEVIC S THEOREM IN IT, 97 4. ELEMENTS OF LOGIC IN
FRAGMENTS 98 (A) ARITHMETIZING PROVABILITY 98 XII TABLE OF CONTENTS (B)
ARITHMETIZING MODEL THEORY 102 (C) APPLICATIONS TO ARITHMETIC 105
CHAPTER II FRAGMENTS AND COMBINATORICS 111 1. RAMSEY S THEOREMS AND
FRAGMENTS 111 (A) STATEMENT OF RESULTS 111 (B) PROOFS (OF 1.5, 1.7, 1.9)
115 (C) PROOFS (OF 1.6, 1.8, 1.10) 118 2. INSTANCES OF THE
PARIS-HARRINGTON PRINCIPLE AND CONSISTENCY STATEMENTS 121 (A)
INTRODUCTION AND STATEMENT OF RESULTS 121 (B) SOME COMBINATORICS 122 (C)
PROOFOF CON (IE + ZV(J7F))-+(P#)* (FOR U 1) . . . . 124 (D) STRONG
INDISCERNIBLES 125 (E) FINAL CONSIDERATIONS 129 3. SCHWICHTENBERG-WAINER
HIERARCHY AND A-LARGE SETS 132 (A) ORDINALS IN IE 133 (B) TRANSFINITE
INDUCTION AND FRAGMENTS 138 (C) A-LARGE SETS IN IS 139 (D)
SCHWICHTENBERG-WAINER HIERARCHY 140 PARTB CHAPTER III SELF-REFERENCE 147
1. PRELIMINARIES 148 (A) INTERPRETABILITY AND PARTIAL CONSERVATIVITY 148
(B) THEORIES CONTAINING ARITHMETIC; SEQUENTIAL THEORIES; PA AND ACA 0
150 (C) NUMERATIONS AND BINUMERATIONS 155 2. SELF-REFERENCE AND GOEDEL S
THEOREMS, REFLEXIVE THEORIES . . . . 158 (A) EXISTENCE OF FIXED POINTS
158 (B) GOEDEL S FIRST INCOMPLETENESS THEOREM AND RELATED TOPICS . . 160
(C) GOEDEL S SECOND INCOMPLETENESS THEOREM 163 (D) PURE EXTENSIONS OF PA
168 (E) INTERPRETABILITY IN PURE EXTENSIONS OF PA 169 3. DEFMABLE CUTS
171 (A) DEFINABLE CUTS AND THEIR PROPERTIES 172 (B) A STRONG FORM OF
GOEDEL S SECOND INCOMPLETENESS THEOREM . 173 (C) HERBRAND PROVABILITY AND
HERBRAND CONSISTENCY 179 (D) CUTS AND INTERPRETATIONS 186 4. PARTIAL
CONSERVATIVITY AND INTERPRETABILITY 189 (A) SOME PROMINENT EXAMPLES 190
TABLE OF CONTENTS XIII (B) GENERAL THEOREMS ON PARTIAL CONSERVATIVITY;
SOME FIXED-POINT THEOREMS 195 (C) APPLICATIONS, MAINLY TO
INTERPRETABILITY 206 CHAPTER IV MODELS OF FRAGMENTS OF ARITHMETIC 213 1.
SOME BASIC CONSTRUCTIONS 214 (A) PRELIMINARIES 214 (B) DEFINABLE
ULTRAPOWER OF THE STANDARD MODEL 216 (C) ON SUBMODELS AND CUTS 218 (D)
MODELS FOR THE HIERARCHY 220 (E) ELEMENTARY END EXTENSIONS 227 (F) A
CONSERVATION RESULT 230 2. CUTS IN MODELS OF ARITHMETIC WITH A TOP 232
(A) ARITHMETIC WITH A TOP AND ITS MODELS 232 (B) CUTS 234 (C)
EXTENDABLE, RESTRAINABLE AND RAMSEY CUTS 236 (D) SATISFACTION IN FINITE
STRUCTURES WITH AN APPLICATION TO MODELS OF IX 1 ! 241 3. PROVABLY
RECURSIVE FUNCTIONS AND THE METHOD OF INDICATORS . . . 245 (A) PROVABLY
RECURSIVE FUNCTIONS, ENVELOPES 245 (B) INDICATORS AND PARIS SEQUENCES
247 (C) PARIS SEQUENCES OF THE FIRST KIND 250 (D) PARIS SEQUENCES OF THE
SECOND KIND 253 (E) FURTHER CONSEQUENCES 257 4. FORMALIZING MODEL THEORY
258 (A) SOME RESULTS ON SATISFACTION AND CONSISTENCY 259 (B) A
CONSERVATION RESULT IN IS 260 (C) APPENDIX: ANOTHER CONSERVATION RESULT
263 PARTC CHAPTER V BOUNDED ARITHMETIC 267 1. A SURVEY OF WEAK FRAGMENTS
OF ARITHMETIC 268 (A) FRAGMENTS OF ARITHMETIC 268 2. A BRIEF
INTRODUCTION TO COMPLEXITY THEORY 276 (A) TIME AND SPACE COMPLEXITY
CLASSES 277 (B) NONDETERMINISTIC COMPUTATIONS 279 (C) DEGREES AND
/VP-COMPLETENESS 280 (D) ORACLE COMPUTATIONS 282 (E) THE LINEAR TIME
HIERARCHY AND THE POLYNOMIAL HIERARCHY . . 283 (F) NEPOMNJASCIJ S
THEOREM 285 (G) THE DIAGONAL METHOD FOR SEPAXATING COMPLEXITY CLASSES .
. 288 XIV TABLE OF CONTENTS 3. EXPONENTIATION, CODING SEQUENCES AND
FORMALIZATION OF SYNTAX IN ISQ 294 (A) INTRODUCTION 294 (B) SETS AND
SEQUENCES 295 (C) THE EXPONENTIATION RELATION 299 (D) DEVELOPING IS 0 +
UE 303 (E) THE NUMBER OF ONES IN A BINARY EXPANSION 304 (F) CODING
SEQUENCES 309 (G) SYNTACTICAL CONCEPTS 312 (H) FORMALIZATIONS BASED ON
CONTEXT-FREE GRAMMARS 315 4. WITNESSING FUNCTIONS 320 (A) INTRODUCTION
320 (B) FRAGMENTS OF BOUNDED ARITHMETIC 320 (C) DEFINABILITY OF TURING
MACHINE COMPUTATIONS IN FRAGMENTS OF BOUNDED ARITHMETIC 330 (D)
WITNESSING FUNCTIONS 337 (E) ON THE FINITE AXIOMATIZABILITY OF BOUNDED
ARITHMETIC . . . 350 5. INTERPRETABILITY AND CONSISTENCY 360 (A)
INTRODUCTION 360 (B) TRUTH DEFINITIONS FOR BOUNDED FORMULAE 361 (C) AN
INTERPRETATION OF IEQ IN Q 366 (D) CUT-ELIMINATION AND HERBRAND S
THEOREM IN BOUNDED ARITHMETIC 371 (E) THE UE 1 THEOREMS OF IS 0 + EXP 380
(F) INCOMPLETENESS THEOREMS 386 (G) ON THE LIMITED USE OF EXPONENTIATION
393 BIBLIOGRAPHICAL REMARKS AND FURTHER READING 397 BIBLIOGRAPHY 409
INDEX OF TERMS 455 INDEX OF SYMBOLS 459
|
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| bvnumber | BV008029056 |
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| discipline | Mathematik |
| format | Book |
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| language | English |
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| physical | XIV, 460 S. |
| publishDate | 1993 |
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| series2 | Perspectives in mathematical logic |
| spelling | Hájek, Petr 1940- Verfasser (DE-588)143699024 aut Metamathematics of first-order arithmetic Petr Hájek ; Pavel Pudlák Berlin u.a. Springer 1993 XIV, 460 S. txt rdacontent n rdamedia nc rdacarrier Perspectives in mathematical logic Literaturverz. S. 410 - 453 Arithmétique - Fondements ram arithmetique bornee inriac fondement arithmetique inriac modele arithmetique inriac theoreme G?de 1 inriac Arithmetic Foundations Peano-Arithmetik (DE-588)4290970-3 gnd rswk-swf Arithmetik (DE-588)4002919-0 gnd rswk-swf Metamathematik (DE-588)4074759-1 gnd rswk-swf Arithmetik (DE-588)4002919-0 s DE-604 Peano-Arithmetik (DE-588)4290970-3 s Metamathematik (DE-588)4074759-1 s 1\p DE-604 Pudlák, Pavel Verfasser aut GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005283251&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Hájek, Petr 1940- Pudlák, Pavel Metamathematics of first-order arithmetic Arithmétique - Fondements ram arithmetique bornee inriac fondement arithmetique inriac modele arithmetique inriac theoreme G?de 1 inriac Arithmetic Foundations Peano-Arithmetik (DE-588)4290970-3 gnd Arithmetik (DE-588)4002919-0 gnd Metamathematik (DE-588)4074759-1 gnd |
| subject_GND | (DE-588)4290970-3 (DE-588)4002919-0 (DE-588)4074759-1 |
| title | Metamathematics of first-order arithmetic |
| title_auth | Metamathematics of first-order arithmetic |
| title_exact_search | Metamathematics of first-order arithmetic |
| title_full | Metamathematics of first-order arithmetic Petr Hájek ; Pavel Pudlák |
| title_fullStr | Metamathematics of first-order arithmetic Petr Hájek ; Pavel Pudlák |
| title_full_unstemmed | Metamathematics of first-order arithmetic Petr Hájek ; Pavel Pudlák |
| title_short | Metamathematics of first-order arithmetic |
| title_sort | metamathematics of first order arithmetic |
| topic | Arithmétique - Fondements ram arithmetique bornee inriac fondement arithmetique inriac modele arithmetique inriac theoreme G?de 1 inriac Arithmetic Foundations Peano-Arithmetik (DE-588)4290970-3 gnd Arithmetik (DE-588)4002919-0 gnd Metamathematik (DE-588)4074759-1 gnd |
| topic_facet | Arithmétique - Fondements arithmetique bornee fondement arithmetique modele arithmetique theoreme G?de 1 Arithmetic Foundations Peano-Arithmetik Arithmetik Metamathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005283251&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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