Metamathematics of first order arithmetic:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin ; Heidelberg ; New York ; Barcelona ; Budapest u.a.
Springer
1998
|
| Ausgabe: | 2. printing |
| Schriftenreihe: | Perspectives in mathematical logic
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 409 - 453 |
| Beschreibung: | XIV, 460 S. |
| ISBN: | 354063648X |
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| 246 | 1 | 0 | |a first-order |
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| 490 | 0 | |a Perspectives in mathematical logic | |
| 500 | |a Literaturverz. S. 409 - 453 | ||
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Datensatz im Suchindex
| _version_ | 1826212324890902528 |
|---|---|
| adam_text |
PETR HAJEK
PAVEL
PUDLAK
METAMATHEMATIC S
OF
FIRST-ORDER
ARITHMETIC
SPRINGER
TABLE OF CONTENTS
INTRODUCTION 1
PRELIMINARIES 5
(A) SOME LOGIC 5
V
(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
IE\
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
IE\
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 ILL
1.
RAMSEY'S THEOREMS AND FRAGMENTS ILL
(A) STATEMENT OF RESULTS ILL
(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) PROOF OF
CON*(IS* +
2V(/7J))
-* (PH)
U
(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
PART B
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 GODEL'S THEOREMS, REFLEXIVE THEORIES .
.
.
. 158
(A)
EXISTENCE OF FIXED POINTS 158
(B) GODEL'S FIRST INCOMPLETENESS THEOREM AND RELATED TOPICS . . 160
(C)
GODEL'S SECOND INCOMPLETENESS THEOREM 163
(D) PURE EXTENSIONS
OI PA
. , 168
(E) INTERPRETABILITY IN PURE EXTENSIONS OF PA 169
3.
DEFINABLE CUTS . . 171
(A) DEFINABLE CUTS AND THEIR PROPERTIES 172
(B) A STRONG FORM OF GODEL'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
IS
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) NONDETENNINISTIC COMPUTATIONS 279
(C) DEGREES AND WP-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 SEPARATING COMPLEXITY CLASSES . . 288
XIV TABLE OF CONTENTS
3.
EXPONENTIATION, CODING SEQUENCES
AND
FORMALIZATION OF SYNTAX IN
IEQ
. 294
(A)
INTRODUCTION 294
(B) SETS AND SEQUENCES 295
(C) THE EXPONENTIATION RELATION 299
(D) DEVELOPING
IEQ
+ Q
X
. 303
(E) THE NUMBER OF ONES IN A BINARY EXPANSION 304
(F) CODING SEQUENCES 309
(G) SYNTACTICAL CONCEPTS YY. 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
YY
371
(E) THE I7I THEOREMS OF
IE
Q
+ 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 YY YY YY 459 |
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| author | Hájek, Petr 1940- Pudlák, Pavel |
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| ctrlnum | (OCoLC)41493548 (DE-599)BVBBV011801667 |
| dewey-full | 513 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 513 - Arithmetic |
| dewey-raw | 513 |
| dewey-search | 513 |
| dewey-sort | 3513 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Philosophie |
| edition | 2. printing |
| format | Book |
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| id | DE-604.BV011801667 |
| illustrated | Not Illustrated |
| indexdate | 2025-03-10T13:03:09Z |
| institution | BVB |
| isbn | 354063648X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007968223 |
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| physical | XIV, 460 S. |
| publishDate | 1998 |
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| publisher | Springer |
| record_format | marc |
| series2 | Perspectives in mathematical logic |
| spelling | Hájek, Petr 1940- Verfasser (DE-588)143699024 aut Metamathematics of first order arithmetic Petr Hájek ; Pavel Pudlák first-order 2. printing Berlin ; Heidelberg ; New York ; Barcelona ; Budapest u.a. Springer 1998 XIV, 460 S. txt rdacontent n rdamedia nc rdacarrier Perspectives in mathematical logic Literaturverz. S. 409 - 453 Arithmétique - Fondements Logica gtt Logik Arithmetic Foundations Arithmetik (DE-588)4002919-0 gnd rswk-swf Metamathematik (DE-588)4074759-1 gnd rswk-swf Peano-Arithmetik (DE-588)4290970-3 gnd rswk-swf Peano-Arithmetik (DE-588)4290970-3 s Metamathematik (DE-588)4074759-1 s DE-604 Arithmetik (DE-588)4002919-0 s 1\p DE-604 Pudlák, Pavel Verfasser aut DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007968223&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 Logica gtt Logik Arithmetic Foundations Arithmetik (DE-588)4002919-0 gnd Metamathematik (DE-588)4074759-1 gnd Peano-Arithmetik (DE-588)4290970-3 gnd |
| subject_GND | (DE-588)4002919-0 (DE-588)4074759-1 (DE-588)4290970-3 |
| title | Metamathematics of first order arithmetic |
| title_alt | first-order |
| 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 Logica gtt Logik Arithmetic Foundations Arithmetik (DE-588)4002919-0 gnd Metamathematik (DE-588)4074759-1 gnd Peano-Arithmetik (DE-588)4290970-3 gnd |
| topic_facet | Arithmétique - Fondements Logica Logik Arithmetic Foundations Arithmetik Metamathematik Peano-Arithmetik |
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