How to measure the infinite: mathematics with infinite and infinitesimal numbers
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo
World Scientific
[2019]
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | xxviii, 317 Seiten |
| ISBN: | 9789812836373 |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1804145822651121664 |
|---|---|
| adam_text | Contents Preface V Plan of the Book XV Notation xvii Historical Introduction Ancient times 1. 2. The rise of calculus 3. The ban of infinitesimals 4. Non-Archimedean mathematics Part 1. Alpha-Calculus xxi xxi xxiii xxvi xxvii 1 Chapter 1. Extending the Real Line 1. Ordered fields 2. Infinitesimal numbers 3. The smallest non-Archimedean field 4. Proper extensions of the real line 5. Standard parts 6. Monads and galaxies 9 11 Chapter 2, Alpha-Calculus I. The axioms of Alpha-Calculus 2. First properties of Alpha-Calculus 3. Hyper-extensions of sets of reals 4. The Alpha-measure and the qualified sets 5. The transfer principle, informally 6. Hyper-extensions of functions 13 14 17 20 23 26 27 JX 3 3 4 6 8
CONTENTS x 7. Some more relevant basic properties 8. Hyper-extensions of sets of numbers 9. The Qualified Set Axiom 10. Rings and ideals 11. Models of Alpha-Calculus 30 32 34 38 41 Chapter 3. Infinitesimal Analysis by Alpha-Calculus 1. The normal forms 2. Infimum and supremum 3. Continuity 4. Uniform continuity 5. Derivatives 6. Limits 7. Alpha-limit versus limit 8. The order of magnitude 9. Hyperfinitely long sums and series 10. The grid integral 11. Equivalences with the “standard” definitions 12. Grid integral versus Riemann integral 13. Remarks and comments 47 47 48 49 52 53 54 57 58 61 63 67 71 72 Part 2. 75 Alpha-Theory Chapter 4. Introducing the Alpha-Theory 1. The axioms of Alpha-Theory 2. First properties of Alpha-Theory 3. Some detailed proofs 4. Hyper-images of sets 5. Hyper-images of functions 6. Functions of several variables 7. Hyperfinite sets 8. Hyperfinite sums 9. Internal objects 10. Remarks and comments 77 77 80 83 88 91 94 96 98 100 107 Chapter 5. Logic and Alpha-Theory 1. Some logic formalism 2. Transfer principle 3. Transfer and internal sets 4. The transfer as a unifying principle 5. Remarks and comments 109 109 115 118 120 122
CONTENTS xi Chapter 6. Complements of Alpha-Theory 1. Overspill and underspill 2. Countable saturation 3. Cauchy infinitesimal principle 4. The S-topology 5. The topology of Alpha-limits 6. Superstructures 7. Models of Alpha-Theory 8. Existence of reflexive Alpha-morphisms 9. Remarks and comments 123 123 126 129 133 137 138 142 146 151 Part 3. 153 Applications Chapter 7. First Applications 1. The real line as a quotient of hyperrationals 2. Ramsey’s Theorem 3. Grid functions 4. Grid differential equations 5. Peano’s theorem 155 155 156 159 163 165 Chapter 8. Gauge Spaces L Main definitions 2. Gauge Abelian spaces 3. Topological notions for gauge spaces 4. Topology theorems in gauge spaces 5. Gauge spaces versus topological spaces 6. The Epsilon-gauges 169 169 172 173 174 176 181 Chapter 9. Gauge Quotients 1. Definition of gauge quotients 2. The differential Epsilon-ring 3. Distributions as a gauge quotient 4. Distributions as functionals 5. Grid functions and distributions 185 185 187 189 191 193 Chapter 10. Stochastic Differential Equations 1. Preliminary remarks on the whitenoise 2. Stochastic grid equations 3. Itô’s formula for grid functions 4. The Fokker-Plank equation 5. Remarks and comments 197 197 199 201 202 206
xii Part 4. CONTENTS Foundations 209 Chapter 11. Ultrafilters and Ultrapowers 1. Filters and ultrafilters 2. Ultrafilters as measures and as ideals 3. Ultrapowers, the basic examples 4. Ultrapowers as models of Alpha-Calculus 211 211 215 218 220 Chapter 12. The Uniqueness Problem 1. Isomorphic models of Alpha-Calculus 2. Equivalent models of Alpha-Calculus 3. Uniqueness up to countable equivalence 223 223 225 227 Chapter 13. Alpha-Theory and Nonstandard Analysis 233 1. Nonstandard analysis, a quick presentation 234 2. Alpha-Theory is more general than nonstandard analysis 236 3. Remarks and comments 238 Chapter 14. Alpha-Theory as a Nonstandard Set Theory 1. The axioms of AST 2. Alpha Set Theory versus ZFC 3. Cauchy infinitesimal principle and special ultrafilters 4. The strength of Cauchy infinitesimal principle 5. The strength of a Hausdorff S-topology 6. Remarks and comments 241 242 247 251 254 257 259 Part 5. 261 Numerosity Theory Chapter 15. Counting Systems 1. The idea of counting 2. Cardinals and ordinals as counting systems 3. Three different ways of counting 4. The equisize relation 263 263 267 270 271 Chapter 16. Alpha-Theory and Numerosity 1. Labelled sets 2. Alpha-numerosity 3. Finite parts and sets of functions 4. Point sets of natural numbers 5. Numerosity of sets of natural numbers 6. Properties of Alpha-numerosity 7. Numerosities of sets of rational numbers 8. Zermelo’s principle and Alpha-numerosity 9. Asymptotic density and numerosity 277 277 280 281 283 285 287 290 293 295
CONTENTS xiii Chapter 17. A General Numerosity Theory for Labelled Sets 1. Definition and first properties 2. From numerosities to Alpha-Calculus 3. Remarks and comments 297 297 300 306 Bibliography 307 Index 313
HOWTO MEASURE THE INFINITE Mathematics with Infinite and Infinitesimal Numbers This book contains an original introduction to the use of infinitesimal and infinite numbers, namely, the AlphaTheory, which can be considered as an alternative approach to nonstandard analysis. The basic principles are presented in an elementary way by using the ordinary language of mathematics; this is to be contrasted with other presentations of nonstandard analysis where technical notions from logic are required since the beginning. Some applications are included and aimed at showing the power of the theory. The book also provides a comprehensive exposition of the Theory of Numerosity, a new way of counting (countable) infinite sets that maintains the ancient Euclid s Principle: The whole is larger than its parts . The book is organized into five parts: Alpha-Calculus, Alpha-Theory, Applications, Foundations, and Numerosity Theory. World Scientific www.worldscientific.com 7081 he 9789812836373
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| any_adam_object | 1 |
| author | Benci, Vieri Di Nasso, Mauro 1963- |
| author_GND | (DE-588)1184827338 |
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| ctrlnum | (OCoLC)699545069 (DE-599)BSZ326526900 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV039109643 |
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| institution | BVB |
| isbn | 9789812836373 |
| language | English |
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| spelling | Benci, Vieri Verfasser aut How to measure the infinite mathematics with infinite and infinitesimal numbers Vieri Benc, Università di Pisa, Italy, Mauro Di Nasso, Università di Pisa, Italy New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo World Scientific [2019] © 2019 xxviii, 317 Seiten txt rdacontent n rdamedia nc rdacarrier Infinitesimalanalysis (DE-588)4161657-1 gnd rswk-swf Transfinite Zahl (DE-588)4471005-7 gnd rswk-swf Nichtarchimedische Analysis (DE-588)4171709-0 gnd rswk-swf Nichtarchimedische Analysis (DE-588)4171709-0 s Infinitesimalanalysis (DE-588)4161657-1 s Transfinite Zahl (DE-588)4471005-7 s DE-604 Di Nasso, Mauro 1963- Verfasser (DE-588)1184827338 aut Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022653390&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022653390&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Benci, Vieri Di Nasso, Mauro 1963- How to measure the infinite mathematics with infinite and infinitesimal numbers Infinitesimalanalysis (DE-588)4161657-1 gnd Transfinite Zahl (DE-588)4471005-7 gnd Nichtarchimedische Analysis (DE-588)4171709-0 gnd |
| subject_GND | (DE-588)4161657-1 (DE-588)4471005-7 (DE-588)4171709-0 |
| title | How to measure the infinite mathematics with infinite and infinitesimal numbers |
| title_auth | How to measure the infinite mathematics with infinite and infinitesimal numbers |
| title_exact_search | How to measure the infinite mathematics with infinite and infinitesimal numbers |
| title_full | How to measure the infinite mathematics with infinite and infinitesimal numbers Vieri Benc, Università di Pisa, Italy, Mauro Di Nasso, Università di Pisa, Italy |
| title_fullStr | How to measure the infinite mathematics with infinite and infinitesimal numbers Vieri Benc, Università di Pisa, Italy, Mauro Di Nasso, Università di Pisa, Italy |
| title_full_unstemmed | How to measure the infinite mathematics with infinite and infinitesimal numbers Vieri Benc, Università di Pisa, Italy, Mauro Di Nasso, Università di Pisa, Italy |
| title_short | How to measure the infinite |
| title_sort | how to measure the infinite mathematics with infinite and infinitesimal numbers |
| title_sub | mathematics with infinite and infinitesimal numbers |
| topic | Infinitesimalanalysis (DE-588)4161657-1 gnd Transfinite Zahl (DE-588)4471005-7 gnd Nichtarchimedische Analysis (DE-588)4171709-0 gnd |
| topic_facet | Infinitesimalanalysis Transfinite Zahl Nichtarchimedische Analysis |
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