Nonstandard analysis, axiomatically:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin ; Heidelberg ; New York
Springer
2004
|
| Schriftenreihe: | Springer monographs in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 388 - 396 |
| Beschreibung: | XVI, 408 S. 24 cm |
| ISBN: | 354022243X |
Internformat
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| 245 | 1 | 0 | |a Nonstandard analysis, axiomatically |c Vladimir Kanovei ; Michael Reeken |
| 264 | 1 | |a Berlin ; Heidelberg ; New York |b Springer |c 2004 | |
| 300 | |a XVI, 408 S. |c 24 cm | ||
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Datensatz im Suchindex
| _version_ | 1804132942441611264 |
|---|---|
| adam_text | VLADIMIR KANOVEI * MICHAEL REEKEN NONSTANDARD ANALYSIS, AXIOMATICALLY 4Y
SPRINGER TABLE OF CONTENTS INTRODUCTION 1 BASIC NOTATION 10 1 GETTING
STARTED 11 1.1 THE AXIOMATICAL SYSTEM OF HRBACEK SET THEORY 12 1.1A THE
UNIVERSE OF HST 12 1.1B AXIOMS FOR THE EXTERNAL UNIVERSE 14 1.1C AXIOMS
FOR STANDARD AND INTERNAL SETS 14 L.LD WELL-FOUNDED SETS 16 L.LE THE
6-STRUCTURE OF INTERNAL AND WELL-FOUNDED SETS 17 L.LF AXIOMS FOR SETS OF
STANDARD SIZE 19 L.LG PUTTING IT ALL TOGETHER 20 L.LH ZERMELO - PRAENKEL
THEORY ZFC 20 1.2 BASIC ELEMENTS OF THE NONSTANDARD UNIVERSE 22 1.2A HOW
TO DEFINE FUNDAMENTAL SET THEORETIC NOTIONS IN HST . . 22 1.2B CLOSURE
PROPERTIES AND ABSOLUTENESS 22 1.2C ORDINALS AND CARDINALS 24 1.2D
NATURAL NUMBERS, FINITE AND *-FMITE SETS 25 1.2E HEREDITARILY FINITE
SETS 28 1.3 SETS OF STANDARD SIZE 29 1.3A CARDINALITIES OF SETS OF
STANDARD SIZE 29 1.3B SATURATION AND THE HRBACEK PARADOX 30 1.3C THE
PRINCIPLE OF EXTENSION 32 1.4 THE CLASS A 34 1.4A BASIC PROPERTIES OF
A 34 1.4B CUTS (INITIAL SEGMENTS) OF *-ORDINALS 35 1.4C MONADS AND
TRANSVERSALS 37 1.4D ON NON-WELL-FOUNDED CARDINALITIES 38 1.4E SMALL AND
LARGE SETS 40 1.5 SOME FINER POINTS 42 1.5A VON NEUMANN HIERARCHY AND
REFLECTION IN ZFC 42 1.5B VON NEUMANN HIERARCHY OVER INTERNAL SETS IN
HST 44 1.5C CLASSES AND STRUCTURES 45 1.5D INTERPRETATIONS 47 1.5E
MODELS 48 1.5F SIMULATION OF MODELS OF ZFC 49 1.5G ASTERISK IS AN
ELEMENTARY EMBEDDING 50 HISTORICAL AND OTHER NOTES TO CHAPTER 1 52 XII
TABLE OF CONTENTS 2 ELEMENTARY REAL ANALYSIS IN THE NONSTANDARD UNIVERSE
53 2.1 HYPERREAL LINE 54 2.1A HYPERREALS 54 2.1B FUNDAMENTALS OF
NONSTANDARD REAL ANALYSIS 56 2.1C DIRECTED SATURATION 57 2.ID
NONSTANDARD CHARACTERIZATION OF CLOSED AND COMPACT SETS .. 58 2.2
SEQUENCES AND FUNCTIONS 59 2.2A LIMITS 60 2.2B CONTINUOUS FUNCTIONS 61
2.2C INTERMEDIATE VALUE THEOREM 62 2.2D ROBINSON S LEMMA AND UNIFORM
LIMITS 62 2.3 TOPICS IN NONSTANDARD REAL ANALYSIS 64 2.3A SHADOWS AND
EQUIVALENCES 64 2.3B NEAR-STANDARD ELEMENTS 66 2.3C TOPOLOGY 69 2.4 TWO
SPECIAL APPLICATIONS 73 2.4A EULER FACTORIZATION OF THE SINE FUNCTION 73
2.4B JORDAN CURVE THEOREM 76 HISTORICAL AND OTHER NOTES TO CHAPTER 2 81
3 THEORIES OF INTERNAL SET S 83 3.1 INTRODUCTION TO INTERNAL SET
THEORIES 84 3.1A INTERNAL SET THEORY 84 3.1B BOUNDED SET THEORY 86 3.1C
INTERNAL SETS INTERPRET BST IN THE EXTERNAL UNIVERSE 87 3.ID BASIC
INTERNAL SET THEORY 88 3.1E STANDARD NATURAL NUMBERS AND STANDARD FINITE
SETS 90 3.IF REMARKS ON BASIC IDEALIZATION AND SATURATION 92 3.2
DEVELOPMENT OF BOUNDED SET THEORY 93 3.2A HALF-BOUNDED FORMS OF
IDEALIZATION 93 3.2B REDUCTION TO TWO EXTERNAL QUANTIFIERS 94 3.2C
FINITE AXIOMATIZABILITY OF BST AND OTHER COROLLARIES 95 3.2D COLLECTION
IN BST 97 3.2E OTHER BASIC THEOREMS OF BST 99 3.2F INTRODUCTION TO THE
PROBLEM OF EXTERNAL SETS 101 3.2G MORE ON EXTERNAL SETS IN BST 104 3.3
INTERNAL THEORIES WITH PARTIAL SATURATION 105 3.3A TWO SCHEMES OF
PARTIALLY SATURATED INTERNAL THEORIES 105 3.3B W-DEEP BASIC IDEALIZATION
SCHEME 106 3.3C K-SIZE BASIC IDEALIZATION SCHEME 109 3.4 DEVELOPMENT OF
NELSON S INTERNAL SET THEORY ILL 3.4A BOUNDED SETS IN 1ST ILL 3.4B
BOUNDED FORMULAS: REDUCTION TO TWO EXTERNAL QUANTIFIERS . 113 3.4C
COLLECTION IN 1ST 114 3.4D UNIQUENESS IN 1ST 117 3.5 TRUTH DEFINITION IN
INTERNAL SET THEORY 118 3.5A TRUTH DEFINITION FOR THE STANDARD UNIVERSE
118 3.5B CONNECTION WITH THE ORDINARY TRUTH 120 3.5C EXTENSION OF THE
DEFINITION OF FORMAL TRUTH 122 TABLE OF CONTENTS XIII 3.6 SECOND EDITION
OF 1ST 124 3.6A STANDARD AND NONSTANDARD THEORIES OF NELSON S SYSTEM
.... 124 3.6B THE BACKGROUND NONSTANDARD UNIVERSE 125 3.6C THREE MYTHS
OF 1ST 127 HISTORICAJ AND OTHER NOTES TO CHAPTER 3 129 METAMATHEMATICS
OF INTERNAL THEORIES 131 4.1 OUTLINE OF METAMATHEMATICAL PROPERTIES 132
4.1A NONSTANDARD EXTENSIONS OF STRUCTURES 132 4.1B NONSTANDARD
EXTENSIONS OF THEORIES 133 4.1C COMMENTS 134 4.ID METAMATHEMATICS OF
INTERNAL THEORIES: THE MAIN RESULTS .... 136 4.2 ULTRAPOWERS AND
SATURATED EXTENSIONS 138 4.2A SATURATED STRUCTURES AND NONSTANDARD SET
THEORIES 138 4.2B QUOTIENT POWER EXTENSIONS 140 4.2C ADEQUATE AND GOOD
ULTRAFILTERS AND ULTRAPOWERS 142 4.2D ELEMENTARY CHAINS OF STRUCTURES
144 4.3 METAMATHEMATICS OF BST 146 4.3A WARMUP: SEVERAL EXAMPLES 146
4.3B INFINITE FUBINI PRODUCTS OF ADEQUATE ULTRAFILTERS 148 4.3C STANDARD
CORE INTERPRETATION OF BST IN ZFC 150 4.3D SATURATED STANDARD CORE
INTERPRETATION 152 4.4 THE CONSERVATIVITY AND EQUICONSISTENCY OF 1S T
154 4.4A GOOD EXTENSIONS OF VON NEUMANN SETS IN ZFC UNIVERSE .. . 154
4.4B ITERATED ADEQUATE EXTENSIONS OF VON NEUMANN SETS 155 4.4C ITERATED
ADEQUATE EXTENSIONS IN THE INVERSION OF ZFC 156 4.4D LONG ITERATED
QUOTIENT POWER CHAINS 156 4.4E CONSERVATIVITY OF 1ST BY INNER MODELS 157
4.5 NON-REDUCIBILITY OF 1ST 159 4.5A THE MINIMALITY AXIOM 159 4.5B THE
SOURCE OF COUNTEREXAMPLES 160 4.5C THE ULTRAFILTER 161 4.5D DEFINABLE
ADEQUATE QUOTIENT POWER 163 4.5E COROLLARIES AND REMARKS 164 4.6
INTERPRETABILITY OF 1ST IN A STANDARD THEORY 166 4.6A STANDARD THEORY
WITH A GLOBAL CHOICE AND A TRUTH PREDICATE 166 4.6B FORMALLY DEFINABLE
CLASSES 168 4.6C A NONSTANDARD THEORY EXTENDING 1ST 169 4.6D THE
ULTRAFILTER 170 4.6E THE INTERPRETATION 173 4.6F EXTENDIBILITY OF
STANDARD MODELS 175 HISTORICAJ AND OTHER NOTES TO CHAPTER 4 176
DEFINABLE EXTERNAL SETS AND METAMATHEMATICS OF HST 179 5.1 INTRODUCTION
TO METAMATHEMATICS OF HST 180 5.1A INTERNAL CORE EMBEDDINGS AND
INTERPRETABILITY 180 5.1B METAMATHEMATICS OF HST : AN OVERVIEW 181 5.2
FROM INTERNAL TO ELEMENTARY EXTERNAL SETS 184 5.2A INTERPRETATION OF
EEST IN BST 184 XIV TABLE OF CONTENTS 5.2B ELEMENTARY EXTERNAL SETS IN
EXTERNAL THEORIES 186 5.2C SOME BASIC THEOREMS OF EEST 188 5.2D STANDARD
SIZE, NATURAL NUMBERS, FINITENESS IN EEST 189 5.3 ASSEMBLING OF EXTERNAL
SETS IN HST 191 5.3A WELL-FOUNDED TREES 191 5.3B CODING OF THE
ASSEMBLING CONSTRUCTION 192 5.3C EXAMPLES OF CODES 193 5.3D REGULAR
CODES 195 5.4 FROM ELEMENTARY EXTERNAL TO ALL EXTERNAL SETS 196 5.4A THE
DOMAIN OF THE INTERPRETATION . 196 5.4B BASIC RELATIONS BETWEEN CODES
198 5.4C THE STRUCTURE OF BASIC RELATIONS 200 5.4D THE INTERPRETATION
AND THE EMBEDDING 202 5.4E VERIFICATION OF THE HST AXIOMS 204 5.4F
SUPERPOSITION OF INTERPRETATIONS 207 5.4G THE PROBLEM OF EXTERNAL SETS
REVISITED 209 5.5 THE CLASS L[B] : SETS CONSTRUCTIBLE FROM INTERNAL SETS
211 5.5A SETS CONSTRUCTIBLE FROM INTERNAL SETS 211 5.5B PROOF OF THE
THEOREM ON I-CONSTRUCTIBLE SETS 212 5.5C THE AXIOM OF B-CONSTRUCTIBILITY
214 5.5D TRANSFINITE CONSTRUCTIONS IN L[I] 215 HISTORICAL AND OTHER
NOTES TO CHAPTER 5 217 6 PARTIALLY SATURATED UNIVERSES AND THE POWER SET
PROBLEM 21 9 6.1 INTERNAL SUBUNIVERSES 220 6.1A SOME BASIC DEFINITIONS
AND RESULTS 220 6.1B RELATIVE STANDARDNESS 221 6.1C SIMPLE RELATIVE
STANDARDNESS 222 6.1D GORDON CLASSES 224 6.1E ASSOCIATED STRUCTURES 225
6.IF MORE ON INTERNAL SUBUNIVERSES 228 6.1G APPENDIX: KUNEN S THEOREM .
. . 229 6.2 PARTIALLY SATURATED INTERNAL UNIVERSES 230 6.2A PARTIALLY
SATURATED CLASSES B K 230 6.2B GOOD INTERNAL SUBUNIVERSES 232 6.2C
INTERNAL UNIVERSES OVER COMPLETE SETS 233 6.3 EXTERNAL UNIVERSES 237
6.3A EXTERNAL UNIVERSES AND INTERNAL CORE EXTENSIONS 237 6.3B VON
NEUMANN CONSTRUCTION OVER NON-TRANSITIVE CLASSES .... 239 6.3C
ABSOLUTENESS FOR EXTERNAL SUBUNIVERSES 240 6.4 PARTIALLY SATURATED
EXTERNAL UNIVERSES 241 6.4A PARTIALLY SATURATED EXTERNAL THEORIES 241
6.4B EXTENSIONS OF THIN CLASSES 243 6.4C CONSTRUCTIBLE EXTENSIONS 244
6.4D CONSTRUCTIBLE EXTENSIONS OF SELF-DEFINABLE CLASSES 246 6.4E THE
CLASSES L[0 K ] 248 6.4F EXTERNAL UNIVERSES OVER COMPLETE SETS 249 6.4G
COLLAPSE ONTO A TRANSITIVE CLASS 251 6.4H OUTLINE OF APPLICATIONS:
SUBUNIVERSES SATISFYING POWER SET. . 252 TABLE OF CONTENTS XV HISTORICAJ
AND OTHER NOTES TO CJIAPTER 6 254 FORCING EXTENSIONS OF THE NONSTANDARD
UNIVERSE 257 7.1 GENERIC EXTENSIONS OF MODELS OF HST 25 8 7.1A GROUND
MODEL 258 7.1B REGULAR EXTENSIONS . 259 7.1C FORCING NOTIONS AND NAMES
260 7.ID ADDING A SET 261 7.1E FORCING RELATION 263 7.IF GENERIC
EXTENSIONS AND THE TRUTH LEMMA 266 7.1G THE EXTENSION MODELS HST 267 7.2
APPLICATIONS: COLLAPSE MAPS AND ISOMORPHISMS 270 7.2A MAKING TWO
INTERNAL SETS EQUINUMEROUS 270 7.2B INTERNAL PRESERVING BIJECTIONS 272
7.2C MAKING ELEMENTARILY EQUIVALENT STRUCTURES ISOMORPHIC 273 7.2D THE
FORCING NOTION 274 7.2E KEY LEMMA 276 7.2F GENERIC ISOMORPHISMS 278 7.3
CONSISTENCY OF THE ISOMORPHISM PROPERTY 279 7.3A THE PRODUCT FORCING
NOTION 280 7.3B EXTERNALIZATION 281 7.3C RESTRICTED FORCING RELATIONS
282 7.3D AUTOMORPHISMS AND THE RESTRICTION PROPERTY 283 7.3E THE PRODUCT
GENERIC EXTENSION 284 HISTORICAL AND OTHER NOTES TO CHAPTER 7 287 OTHER
NONSTANDARD THEORIES 289 8.1 NONSTANDARD SET THEORY OF KAWAI 290 8.1A
THE AXIOMS OF HAWAII S THEORY 290 8.1B METAMATHEMATICAL PROPERTIES 292
8.1C SPECIAL MODEL AXIOM 293 8.2 NONSTANDARD SET THEORY OF HRBACEK 295
8.2A AXIOMS 295 8.2B ADDITIONAL AXIOMS OF COLLECTION 297 8.2C
CONSERVATIVITY AND CONSISTENCY 298 8.2D REMARKS AND EXERCISES 301 8.3
NON-WELL-FOUNDED SET THEORIES 303 8.3A BOFFA S NON-WELL-FOUNDED SET
THEORY 303 8.3B EXTENSIONS OF PROPER CLASSES 305 8.3C APPLICATIONS TO
NONSTANDARD ANALYSIS 306 8.3D ALPHA THEORY 307 8.3E INTERPRETATION OF
ALPHA THEORY IN ZFBC 311 8.4 MISCELLANEA: SOME OTHER THEORIES 312 8.4A A
THEORY WITH DEFINABLE SATURATION 312 8.4B STRATIFIED NONSTANDARD SET
THEORIES 313 8.4C NONSTANDARD CLASS THEORIES 314 HISTORICAL AND OTHER
NOTES TO CHAPTER 8 315 XVI TABLE OF CONTENTS 9 HYPERFINITE DESCRIPTIVE
SET THEORY 317 9.1 INTRODUCTION TO HYPERFINITE DST 319 9.1A GENERAL
SET-UP 319 9.1B COMMENTS ON NOTATION 320 9.1C BOREL AND PROJECTIVE SETS
IN A NONSTANDARD DOMAIN 321 9.ID SOME APPLICATIONS OF COUNTABLE
SATURATION 323 9.1E OPERATION A AND SOUSLIN SETS 324 9.2 OPERATIONS,
COUNTABLY DETERMINED SETS, SHADOWS 325 9.2A OPERATIONS AND QUANTIFIERS
325 9.2B COUNTABLY DETERMINED SETS 327 9.2C SHADOWS OR STANDARD PART
MAPS 329 9.3 STRUCTURE OF THE HIERARCHIES 331 9.3A OPERATIONS ASSOCIATED
WITH BOREL AND PROJECTIVE CLASSES .. . 331 9.3B THE SHADOW THEOREM 332
9.3C CLOSURE PROPERTIES OF THE CLASSES 335 9.4 SOME CLASSICAL QUESTIONS
338 9.4A SEPARATION AND REDUCTION 338 9.4B COUNTABLY DETERMINED SETS
WITH COUNTABLE CROSS-SECTIONS . . 340 9.4C COUNTABLY DETERMINED SETS
WITH INTERNAL AND S? CROSS-SECTIONS 343 9.4D UNIFORMIZATION 344 9.4E
VARIATIONS ON LOUVEAU S THEME 347 9.4F ON SETS WITH II? CROSS-SECTIONS
350 9.5 LOEB MEASURES 351 9.5A DEFINITIONS AND EXAMPLES * 351 9.5B LOEB
MEASURABILITY OF PROJECTIVE SETS 353 9.5C APPROXIMATIONS ALMOST
EVERYWHERE 354 9.5D RANDOMNESS IN A HYPERFINITE DOMAIN 356 9.5E LAW OF
LARGE NUMBERS 358 9.5F RANDOM SEQUENCES AND HYPERFINITE GAMBLING 359 9.6
BOREL AND COUNTABLY DETERMINED CARDINALITIES 362 9.6A PRELIMINARIES 362
9.6B BOREL CARDINALS AND CUTS 364 9.6C PROOF OF THE THEOREM ON BOREL
CARDINALITIES 366 9.6D COMPLETE CLASSIFICATION OF BOREL CARDINALITIES
368 9.6E COUNTABLY DETERMINED CARDINALITIES 368 9.7 EQUIVALENCE
RELATIONS AND QUOTIENTS 370 9.7A SILVER S THEOREM FOR COUNTABLY
DETERMINED RELATIONS 371 9.7B APPLICATION: NONSTANDARD PARTITION
CALCULUS 373 9.7C GENERALIZATION 375 9.7D TRANSVERSALS OF COUNTABLE
EQUIVALENCE RELATIONS 376 9.7E EQUIVALENCE RELATIONS OF MONAD PARTITIONS
378 9.7F BOREL AND COUNTABLY DETERMINED REDUCIBILITY 380 9.7G
REDUCIBILITY STRUCTURE OF MONAD PARTITIONS 382 HISTORICAL AND OTHER
NOTES TO CHAPTER 9 386 REFERENCES 389 INDEX 397
|
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| ctrlnum | (OCoLC)249469375 (DE-599)BVBBV019584586 |
| dewey-full | 515 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515 |
| dewey-search | 515 |
| dewey-sort | 3515 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV019584586 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T20:00:50Z |
| institution | BVB |
| isbn | 354022243X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-012921510 |
| oclc_num | 249469375 |
| open_access_boolean | |
| owner | DE-703 DE-824 DE-384 DE-634 DE-11 DE-188 |
| owner_facet | DE-703 DE-824 DE-384 DE-634 DE-11 DE-188 |
| physical | XVI, 408 S. 24 cm |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer monographs in mathematics |
| spelling | Kanovej, Vladimir Verfasser aut Nonstandard analysis, axiomatically Vladimir Kanovei ; Michael Reeken Berlin ; Heidelberg ; New York Springer 2004 XVI, 408 S. 24 cm txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Literaturverz. S. 388 - 396 Nonstandard-Analysis - Axiomatik Nonstandard mathematical analysis Nonstandard-Analysis (DE-588)4137021-1 gnd rswk-swf Axiomatik (DE-588)4004038-0 gnd rswk-swf Nonstandard-Analysis (DE-588)4137021-1 s Axiomatik (DE-588)4004038-0 s DE-604 Reeken, Michael Verfasser (DE-588)110002253 aut HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012921510&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kanovej, Vladimir Reeken, Michael Nonstandard analysis, axiomatically Nonstandard-Analysis - Axiomatik Nonstandard mathematical analysis Nonstandard-Analysis (DE-588)4137021-1 gnd Axiomatik (DE-588)4004038-0 gnd |
| subject_GND | (DE-588)4137021-1 (DE-588)4004038-0 |
| title | Nonstandard analysis, axiomatically |
| title_auth | Nonstandard analysis, axiomatically |
| title_exact_search | Nonstandard analysis, axiomatically |
| title_full | Nonstandard analysis, axiomatically Vladimir Kanovei ; Michael Reeken |
| title_fullStr | Nonstandard analysis, axiomatically Vladimir Kanovei ; Michael Reeken |
| title_full_unstemmed | Nonstandard analysis, axiomatically Vladimir Kanovei ; Michael Reeken |
| title_short | Nonstandard analysis, axiomatically |
| title_sort | nonstandard analysis axiomatically |
| topic | Nonstandard-Analysis - Axiomatik Nonstandard mathematical analysis Nonstandard-Analysis (DE-588)4137021-1 gnd Axiomatik (DE-588)4004038-0 gnd |
| topic_facet | Nonstandard-Analysis - Axiomatik Nonstandard mathematical analysis Nonstandard-Analysis Axiomatik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012921510&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT kanovejvladimir nonstandardanalysisaxiomatically AT reekenmichael nonstandardanalysisaxiomatically |