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Ⅰ Foundations1

1 What Are the Hyperreals?3

1.1 Infinitely Small and Large3

1.2 Historical Background4

1.3 What Is a Real Number?11

1.4 Historical References14

2 Large Sets15

2.1 Infinitesimals as Variable Quantities15

2.2 Largeness16

2.3 Filters18

2.4 Examples of Filters18

2.5 Facts About Filters19

2.6 Zorn's Lemma19

2.7 Exercises on Filters21

3 Ultrapower Construction of the Hyperreals23

3.1 The Ring of Real-Valued Sequences23

3.2 Equivalence Modulo an Ultrafilter24

3.3 Exercises on Almost-Everywhere Agreement24

3.4 A Suggestive Logical Notation24

3.5 Exercises on Statement Values25

3.6 The Ultrapower25

3.7 Including the Reals in the Hyperreals27

3.8 Infinitesimals and Unlimited Numbers27

3.9 Enlarging Sets28

3.10 Exercises on Enlargement29

3.11 Extending Functions30

3.12 Exercises on Extensions30

3.13 Partial Functions and Hypersequences31

3.14 Enlarging Relations31

3.15 Exercises on Enlarged Relations32

3.16 Is the Hyperreal System Unique?33

4 The Transfer Principle35

4.1 Transforming Statements35

4.2 Relational Structures38

4.3 The Language of a Relational Structure38

4.4 ＊-Transforms42

4.5 The Transfer Principle44

4.6 Justifying Transfer46

4.7 Extending Transfer47

5 Hyperreals Great and Small49

5.1 (Un)limited,Infinitesimal,and Appreciable Numbers49

5.2 Arithmetic of Hyperreals50

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

5.4 Halos,Galaxies,and Real Comparisons52

5.5 Exercises on Halos and Galaxies52

5.6 Shadows53

5.7 Exercises on Infinite Closeness54

5.8 Shadows and Completeness54

5.9 Exercise on Dedekind Completeness55

5.10 The Hypernaturals56

5.11 Exercises on Hyperintegers and Primes57

5.12 On the Existence of Infinitely Many Primes57

Ⅱ Basic Analysis59

6 Convergence of Sequences and Series61

6.1 Convergence61

6.2 Monotone Convergence62

6.3 Limits63

6.4 Boundedness and Divergence64

6.5 Cauchy Sequences65

6.6 Cluster Points66

6.7 Exercises on Limits and Cluster Points66

6.8 Limits Superior and Inferior67

6.9 Exercises on limsup and liminf70

6.10 Series71

6.11 Exercises on Convergence of Series71

7 Continuous Functions75

7.1 Cauchy's Account of Continuity75

7.2 Continuity of the Sine Function77

7.3 Limits of Functions78

7.4 Exercises on Limits78

7.5 The Intermediate Value Theorem79

7.6 The Extreme Value Theorem80

7.7 Uniform Continuity81

7.8 Exercises on Uniform Continuity82

7.9 Contraction Mappings and Fixed Points82

7.10 A First Look at Permanence84

7.11 Exercises on Permanence of Functions85

7.12 Sequences of Functions86

7.13 Continuity of a Uniform Limit87

7.14 Continuity in the Extended Hypersequence88

7.15 Was Cauchy Right?90

8 Differentiation91

8.1 The Derivative91

8.2 Increments and Differentials92

8.3 Rules for Derivatives94

8.4 Chain Rule94

8.5 Critical Point Theorem95

8.6 Inverse Function Theorem96

8.7 Partial Derivatives97

8.8 Exercises on Partial Derivatives100

8.9 Taylor Series100

8.10 Incremental Approximation by Taylor's Formula102

8.11 Extending the Incremental Equation103

8.12 Exercises on Increments and Derivatives104

9 The Riemann Integral105

9.1 Riemann Sums105

9.2 The Integral as the Shadow of Riemann Sums108

9.3 Standard Properties of the Integral110

9.4 Differentiating the Area Function111

9.5 Exercise on Average Function Values112

10 Topology of the Reals113

10.1 Interior,Closure,and Limit Points113

10.2 Open and Closed Sets115

10.3 Compactness116

10.4 Compactness and(Uniform)Continuity119

10.5 Topologies on the Hyperreals120

Ⅲ Internal and External Entities123

11 Internal and External Sets125

11.1 Internal Sets125

11.2 Algebra of Internal Sets127

11.3 Internal Least Number Principle and Induction128

11.4 The Overflow Principle129

11.5 Internal Order-Completeness130

11.6 External Sets131

11.7 Defining Internal Sets133

11.8 The Underflow Principle136

11.9 Internal Sets and Permanence137

11.10 Saturation of Internal Sets138

11.11 Saturation Creates Nonstandard Entities140

11.12 The Size of an Internal Set141

11.13 Closure of the Shadow of an Internal Set142

11.14 Interval Topology and Hyper-Open Sets143

12 Internal Functions and Hyperfinite Sets147

12.1 Internal Functions147

12.2 Exercises on Properties of Internal Functions148

12.3 Hyperfinite Sets149

12.4 Exercises on Hyperfiniteness150

12.5 Counting a Hyperfinite Set151

12.6 Hyperfinite Pigeonhole Principle151

12.7 Integrals as Hyperfinite Sums152

Ⅳ Nonstandard Frameworks155

13 Universes and Frameworks157

13.1 What Do We Need in the Mathematical World?158

13.2 Pairs Are Enough159

13.3 Actually,Sets Are Enough160

13.4 Strong Transitivity161

13.5 Universes162

13.6 Superstructures164

13.7 The Language of a Universe166

13.8 Nonstandard Frameworks168

13.9 Standard Entities170

13.10 Internal Entities172

13.11 Closure Properties of Internal Sets173

13.12 Transformed Power Sets174

13.13 Exercises on Internal Sets and Functions176

13.14 External Images Are External176

13.15 Internal Set Definition Principle177

13.16 Internal Function Definition Principle178

13.17 Hyperfiniteness178

13.18 Exercises on Hyperfinite Sets and Sizes180

13.19 Hyperfinite Summation180

13.20 Exercises on Hyperfinite Sums181

14 The Existence of Nonstandard Entities183

14.1 Enlargements183

14.2 Concurrence and Hyperfinite Approximation185

14.3 Enlargements as Ultrapowers187

14.4 Exercises on the Ultrapower Construction189

15 Permanence,Comprehensiveness,Saturation191

15.1 Permanence Principles191

15.2 Robinson's Sequential Lemma193

15.3 Uniformly Converging Sequences of Functions193

15.4 Comprehensiveness195

15.5 Saturation198

Ⅴ Applications201

16 Loeb Measure203

16.1 Rings and Algebras204

16.2 Measures206

16.3 Outer Measures208

16.4 Lebesgue Measure210

16.5 Loeb Measures210

16.6 μ-Approximability212

16.7 Loeb Measure as Approximability214

16.8 Lebesgue Measure via Loeb Measure215

17 Ramsey Theory221

17.1 Colourings and Monochromatic Sets221

17.2 A Nonstandard Approach223

17.3 Proving Ramsey's Theorem224

17.4 The Finite Ramsey Theorem227

17.5 The Paris-Harrington Version228

17.6 Reference229

18 Completion by Enlargement231

18.1 Completing the Rationals231

18.2 Metric Space Completion233

18.3 Nonstandard Hulls234

18.4 p-adic Integers237

18.5 p-adic Numbers245

18.6 Power Series249

18.7 Hyperfinite Expansions in Base p255

18.8 Exercises257

19 Hyperflnite Approximation259

19.1 Colourings and Graphs260

19.2 Boolean Algebras262

19.3 Atomic Algebras265

19.4 Hyperfinite Approximating Algebras267

19.5 Exercises on Generation of Algebras269

19.6 Connecting with the Stone Representation269

19.7 Exercises on Filters and Lattices272

19.8 Hyperfinite-Dimensional Vector Spaces273

19.9 Exercises on(Hyper)Real Subspaces275

19.10 The Hahn-Banach Theorem275

19.11 Exercises on(Hyper)Linear Functionals278

20 Books on Nonstandard Analysis279

Index283