超实讲义 英文PDF电子书下载

Ⅰ Foundations 1

1 What Are the Hyperreals? 3

1.1 Infinitely Small and Large 3

1.2 Historical Background 4

1.3 What Is a Real Number? 11

1.4 Historical References 14

2 Large Sets 15

2.1 Infinitesimals as Variable Quantities 15

2.2 Largeness 16

2.3 Filters 18

2.4 Examples of Filters 18

2.5 Facts About Filters 19

2.6 Zorn's Lemma 19

2.7 Exercises on Filters 21

3 Ultrapower Construction of the Hyperreals 23

3.1 The Ring of Real-Valued Sequences 23

3.2 Equivalence Modulo an Ultrafilter 24

3.3 Exercises on Almost-Everywhere Agreement 24

3.4 A Suggestive Logical Notation 24

3.5 Exercises on Statement Values 25

3.6 The Ultrapower 25

3.7 Including the Reals in the Hyperreals 27

3.8 Infinitesimals and Unlimited Numbers 27

3.9 Enlarging Sets 28

3.10 Exercises on Enlargement 29

3.11 Extending Functions 30

3.12 Exercises on Extensions 30

3.13 Partial Functions and Hypersequences 31

3.14 Enlarging Relations 31

3.15 Exercises on Enlarged Relations 32

3.16 Is the Hyperreal System Unique? 33

4 The Transfer Principle 35

4.1 Transforming Statements 35

4.2 Relational Structures 38

4.3 The Language of a Relational Structure 38

4.4 ＊-Transforms 42

4.5 The Transfer Principle 44

4.6 Justifying Transfer 46

4.7 Extending Transfer 47

5 Hyperreals Great and Small 49

5.1 (Un)limited,Infinitesimal,and Appreciable Numbers 49

5.2 Arithmetic of Hyperreals 50

5.3 On the Use of"Finite"and"Infinite" 51

5.4 Halos,Galaxies,and Real Comparisons 52

5.5 Exercises on Halos and Galaxies 52

5.6 Shadows 53

5.7 Exercises on Infinite Closeness 54

5.8 Shadows and Completeness 54

5.9 Exercise on Dedekind Completeness 55

5.10 The Hypernaturals 56

5.11 Exercises on Hyperintegers and Primes 57

5.12 On the Existence of Infinitely Many Primes 57

Ⅱ Basic Analysis 59

6 Convergence of Sequences and Series 61

6.1 Convergence 61

6.2 Monotone Convergence 62

6.3 Limits 63

6.4 Boundedness and Divergence 64

6.5 Cauchy Sequences 65

6.6 Cluster Points 66

6.7 Exercises on Limits and Cluster Points 66

6.8 Limits Superior and Inferior 67

6.9 Exercises on limsup and liminf 70

6.10 Series 71

6.11 Exercises on Convergence of Series 71

7 Continuous Functions 75

7.1 Cauchy's Account of Continuity 75

7.2 Continuity of the Sine Function 77

7.3 Limits of Functions 78

7.4 Exercises on Limits 78

7.5 The Intermediate Value Theorem 79

7.6 The Extreme Value Theorem 80

7.7 Uniform Continuity 81

7.8 Exercises on Uniform Continuity 82

7.9 Contraction Mappings and Fixed Points 82

7.10 A First Look at Permanence 84

7.11 Exercises on Permanence of Functions 85

7.12 Sequences of Functions 86

7.13 Continuity of a Uniform Limit 87

7.14 Continuity in the Extended Hypersequence 88

7.15 Was Cauchy Right? 90

8 Differentiation 91

8.1 The Derivative 91

8.2 Increments and Differentials 92

8.3 Rules for Derivatives 94

8.4 Chain Rule 94

8.5 Critical Point Theorem 95

8.6 Inverse Function Theorem 96

8.7 Partial Derivatives 97

8.8 Exercises on Partial Derivatives 100

8.9 Taylor Series 100

8.10 Incremental Approximation by Taylor's Formula 102

8.11 Extending the Incremental Equation 103

8.12 Exercises on Increments and Derivatives 104

9 The Riemann Integral 105

9.1 Riemann Sums 105

9.2 The Integral as the Shadow of Riemann Sums 108

9.3 Standard Properties of the Integral 110

9.4 Differentiating the Area Function 111

9.5 Exercise on Average Function Values 112

10 Topology of the Reals 113

10.1 Interior,Closure,and Limit Points 113

10.2 Open and Closed Sets 115

10.3 Compactness 116

10.4 Compactness and(Uniform)Continuity 119

10.5 Topologies on the Hyperreals 120

Ⅲ Internal and External Entities 123

11 Internal and External Sets 125

11.1 Internal Sets 125

11.2 Algebra of Internal Sets 127

11.3 Internal Least Number Principle and Induction 128

11.4 The Overflow Principle 129

11.5 Internal Order-Completeness 130

11.6 External Sets 131

11.7 Defining Internal Sets 133

11.8 The Underflow Principle 136

11.9 Internal Sets and Permanence 137

11.10 Saturation of Internal Sets 138

11.11 Saturation Creates Nonstandard Entities 140

11.12 The Size of an Internal Set 141

11.13 Closure of the Shadow of an Internal Set 142

11.14 Interval Topology and Hyper-Open Sets 143

12 Internal Functions and Hyperfinite Sets 147

12.1 Internal Functions 147

12.2 Exercises on Properties of Internal Functions 148

12.3 Hyperfinite Sets 149

12.4 Exercises on Hyperfiniteness 150

12.5 Counting a Hyperfinite Set 151

12.6 Hyperfinite Pigeonhole Principle 151

12.7 Integrals as Hyperfinite Sums 152

Ⅳ Nonstandard Frameworks 155

13 Universes and Frameworks 157

13.1 What Do We Need in the Mathematical World? 158

13.2 Pairs Are Enough 159

13.3 Actually,Sets Are Enough 160

13.4 Strong Transitivity 161

13.5 Universes 162

13.6 Superstructures 164

13.7 The Language of a Universe 166

13.8 Nonstandard Frameworks 168

13.9 Standard Entities 170

13.10 Internal Entities 172

13.11 Closure Properties of Internal Sets 173

13.12 Transformed Power Sets 174

13.13 Exercises on Internal Sets and Functions 176

13.14 External Images Are External 176

13.15 Internal Set Definition Principle 177

13.16 Internal Function Definition Principle 178

13.17 Hyperfiniteness 178

13.18 Exercises on Hyperfinite Sets and Sizes 180

13.19 Hyperfinite Summation 180

13.20 Exercises on Hyperfinite Sums 181

14 The Existence of Nonstandard Entities 183

14.1 Enlargements 183

14.2 Concurrence and Hyperfinite Approximation 185

14.3 Enlargements as Ultrapowers 187

14.4 Exercises on the Ultrapower Construction 189

15 Permanence,Comprehensiveness,Saturation 191

15.1 Permanence Principles 191

15.2 Robinson's Sequential Lemma 193

15.3 Uniformly Converging Sequences of Functions 193

15.4 Comprehensiveness 195

15.5 Saturation 198

Ⅴ Applications 201

16 Loeb Measure 203

16.1 Rings and Algebras 204

16.2 Measures 206

16.3 Outer Measures 208

16.4 Lebesgue Measure 210

16.5 Loeb Measures 210

16.6 μ-Approximability 212

16.7 Loeb Measure as Approximability 214

16.8 Lebesgue Measure via Loeb Measure 215

17 Ramsey Theory 221

17.1 Colourings and Monochromatic Sets 221

17.2 A Nonstandard Approach 223

17.3 Proving Ramsey's Theorem 224

17.4 The Finite Ramsey Theorem 227

17.5 The Paris-Harrington Version 228

17.6 Reference 229

18 Completion by Enlargement 231

18.1 Completing the Rationals 231

18.2 Metric Space Completion 233

18.3 Nonstandard Hulls 234

18.4 p-adic Integers 237

18.5 p-adic Numbers 245

18.6 Power Series 249

18.7 Hyperfinite Expansions in Base p 255

18.8 Exercises 257

19 Hyperflnite Approximation 259

19.1 Colourings and Graphs 260

19.2 Boolean Algebras 262

19.3 Atomic Algebras 265

19.4 Hyperfinite Approximating Algebras 267

19.5 Exercises on Generation of Algebras 269

19.6 Connecting with the Stone Representation 269

19.7 Exercises on Filters and Lattices 272

19.8 Hyperfinite-Dimensional Vector Spaces 273

19.9 Exercises on(Hyper)Real Subspaces 275

19.10 The Hahn-Banach Theorem 275

19.11 Exercises on(Hyper)Linear Functionals 278

20 Books on Nonstandard Analysis 279

Index 283