Lectures on the Hyperreals : An Introduction to Nonstandard Analysis

I Foundations 1(59) What Are the Hyperreals? 3(12) Infinitely Small and Large 3(1) Historical Background 4(7) What Is a Real Number? 11(3) Historical References 14(1) Large Sets 15(8) Infinitesimals as Variable Quantities 15(1) Largeness 16(2) Filters 18(1) Examples of Filters 18(1) Facts About Filters 19(1) Zorn's Lemma 19(2) Exercises on Filters 21(2) Ultrapower Construction of the Hyperreals 23(12) The Ring of Real-Valued Sequences 23(1) Equivalence Modulo an Ultrafilter 24(1) Exercises on Almost-Everywhere Agreement 24(1) A Suggestive Logical Notation 24(1) Exercises on Statement Values 25(1) The Ultrapower 25(2) Including the Reals in the Hyperreals 27(1) Infinitesimals and Unlimited Numbers 27(1) Enlarging Sets 28(1) Exercises on Enlargement 29(1) Extending Functions 30(1) Exercises on Extensions 30(1) Partial Functions and Hypersequences 31(1) Enlarging Relations 31(1) Exercises on Enlarged Relations 32(1) Is the Hyperreal System Unique? 33(2) The Transfer Principle 35(14) Transforming Statements 35(3) Relational Structures 38(1) The Language of a Relational Structure 38(4) *-Transforms 42(2) The Transfer Principle 44(2) Justifying Transfer 46(1) Extending Transfer 47(2) Hyperreals Great and Small 49(10) (Un)limited, Infinitesimal, and Appreciable Numbers 49(1) Arithmetic of Hyperreals 50(1) On the Use of ``Finite'' and ``Infinite'' 51(1) Halos, Galaxies, and Real Comparisons 52(1) Exercises on Halos and Galaxies 52(1) Shadows 53(1) Exercises on Infinite Closeness 54(1) Shadows and Completeness 54(1) Exercise on Dedekind Completeness 55(1) The Hypernaturals 56(1) Exercises on Hyperintegers and Primes 57(1) On the Existence of Infinitely Many Primes 57(2) II Basic Analysis 59(64) Convergence of Sequences and Series 61(14) Convergence 61(1) Monotone Convergence 62(1) Limits 63(1) Boundedness and Divergence 64(1) Cauchy Sequences 65(1) Cluster Points 66(1) Exercises on Limits and Cluster Points 66(1) Limits Superior and Inferior 67(3) Exercises on lim sup and lim inf 70(1) Series 71(1) Exercises on Convergence of Series 71(4) Continuous Functions 75(16) Cauchy's Account of Continuity 75(2) Continuity of the Sine Function 77(1) Limits of Functions 78(1) Exercises on Limits 78(1) The Intermediate Value Theorem 79(1) The Extreme Value Theorem 80(1) Uniform Continuity 81(1) Exercises on Uniform Continuity 82(1) Contraction Mappings and Fixed Points 82(2) A First Look at Permanence 84(1) Exercises on Permanence of Functions 85(1) Sequences of Functions 86(1) Continuity of a Uniform Limit 87(1) Continuity in the Extended Hypersequence 88(2) Was Cauchy Right? 90(1) Differentiation 91(14) The Derivative 91(1) Increments and Differentials 92(2) Rules for Derivatives 94(1) Chain Rule 94(1) Critical Point Theorem 95(1) Inverse Function Theorem 96(1) Partial Derivatives 97(3) Exercises on Partial Derivatives 100(1) Taylor Series 100(2) Incremental Approximation by Taylor's Formula 102(1) Extending the Incremental Equation 103(1) Exercises on Increments and Derivatives 104(1) The Riemann Integral 105(8) Riemann Sums 105(3) The Integral as the Shadow of Riemann Sums 108(2) Standard Properties of the Integral 110(1) Differentiating the Area Function 111(1) Exercise on Average Function Values 112(1) Topology of the Reals 113(10) Interior, Closure, and Limit Points 113(2) Open and Closed Sets 115(1) Compactness 116(3) Compactness and (Uniform) Continuity 119(1) Topologies on the Hyperreals 120(3) III Internal and External Entities 123(32) Internal and External Sets 125(22) Internal Sets 125(2) Algebra of Internal Sets 127(1) Internal Least Number Principle and Induction 128(1) The Overflow Principle 129(1) Internal Order-Completeness 130(1) External Sets 131(2) Defining Internal Sets 133(3) The Underflow Principle 136(1) Internal Sets and Permanence 137(1) Saturation of Internal Sets 138(2) Saturation Creates Nonstandard Entities 140(1) The Size of an Internal Set 141(1) Closure of the Shadow of an Internal Set 142(1) Interval Topology and Hyper-Open Sets 143(4) Internal Functions and Hyperfinite Sets 147(8) Internal Functions 147(1) Exercises on Properties of Internal Functions 148(1) Hyperfinite Sets 149(1) Exercises on Hyperfiniteness 150(1) Counting a Hyperfinite Set 151(1) Hyperfinite Pigeonhole Principle 151(1) Integrals as Hyperfinite Sums 152(3) IV Nonstandard Frameworks 155(46) Universes and Frameworks 157(26) What Do We Need in the Mathematical World? 158(1) Pairs Are Enough 159(1) Actually, Sets Are Enough 160(1) Strong Transitivity 161(1) Universes 162(2) Superstructures 164(2) The Language of a Universe 166(2) Nonstandard Frameworks 168(2) Standard Entities 170(2) Internal Entities 172(1) Closure Properties of Internal Sets 173(1) Transformed Power Sets 174(2) Exercises on Internal Sets and Functions 176(1) External Images Are External 176(1) Internal Set Definition Principle 177(1) Internal Function Definition Principle 178(1) Hyperfiniteness 178(2) Exercises on Hyperfinite Sets and Sizes 180(1) Hyperfinite Summation 180(1) Exercises on Hyperfinite Sums 181(2) The Existence of Nonstandard Entities 183(8) Enlargements 183(2) Concurrence and Hyperfinite Approximation 185(2) Enlargements as Ultrapowers 187(2) Exercises on the Ultrapower Construction 189(2) Permanence, Comprehensiveness, Saturation 191(10) Permanence Principles 191(2) Robinson's Sequential Lemma 193(1) Uniformly Converging Sequences of Functions 193(2) Comprehensiveness 195(3) Saturation 198(3) V Applications 201(82) Loeb Measure 203(18) Rings and Algebras 204(2) Measures 206(2) Outer Measures 208(2) Lebesgue Measure 210(1) Loeb Measures 210(2) μ-Approximability 212(2) Loeb Measure as Approximability 214(1) Lebesgue Measure via Loeb Measure 215(6) Ramsey Theory 221(10) Colourings and Monochromatic Sets 221(2) A Nonstandard Approach 223(1) Proving Ramsey's Theorem 224(3) The Finite Ramsey Theorem 227(1) The Paris-Harrington Version 228(1) Reference 229(2) Completion by Enlargement 231(28) Completing the Rationals 231(2) Metric Space Completion 233(1) Nonstandard Hulls 234(3) p-adic Integers 237(8) p-adic Numbers 245(4) Power Series 249(6) Hyperfinite Expansions in Base p 255(2) Exercises 257(2) Hyperfinite Approximation 259(20) Colourings and Graphs 260(2) Boolean Algebras 262(3) Atomic Algebras 265(2) Hyperfinite Approximating Algebras 267(2) Exercises on Generation of Algebras 269(1) Connecting with the Stone Representation 269(3) Exercises on Filters and Lattices 272(1) Hyperfinite-Dimensional Vector Spaces 273(2) Exercises on (Hyper) Real Subspaces 275(1) The Hahn-Banach Theorem 275(3) Exercises on (Hyper) Linear Functionals 278(1) Books on Nonstandard Analysis 279(4) Index 283