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1.4 Historical References 14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