Lectures on the Hyperreals

I Foundations.- 1 What Are the Hyperreals?.- 1.1 Infinitely Small and Large.- 1.2 Historical Background.- 1.3 What Is a Real Number?.- 1.4 Historical References.- 2 Large Sets.- 2.1 Infinitesimals as Variable Quantities.- 2.2 Largeness.- 2.3 Filters.- 2.4 Examples of Filters.- 2.5 Facts About Filters.- 2.6 Zorn’s Lemma.- 2.7 Exercises on Filters.- 3 Ultrapower Construction of the Hyperreals.- 3.1 The Ring of Real-Valued Sequences.- 3.2 Equivalence Modulo an Ultrafilter.- 3.3 Exercises on Almost-Everywhere Agreement.- 3.4 A Suggestive Logical Notation.- 3.5 Exercises on Statement Values.- 3.6 The Ultrapower.- 3.7 Including the Reals in the Hyperreals.- 3.8 Infinitesimals and Unlimited Numbers.- 3.9 Enlarging Sets.- 3.10 Exercises on Enlargement.- 3.11 Extending Functions.- 3.12 Exercises on Extensions.- 3.13 Partial Functions and Hypersequences.- 3.14 Enlarging Relations.- 3.15 Exercises on Enlarged Relations.- 3.16 Is the Hyperreal System Unique?.- 4 The Transfer Principle.- 4.1 Transforming Statements.- 4.2 Relational Structures.- 4.3 The Language of a Relational Structure.- 4.4 *-Transforms.- 4.5 The Transfer Principle.- 4.6 Justifying Transfer.- 4.7 Extending Transfer.- 5 Hyperreals Great and Small.- 5.1 (Un)limited, Infinitesimal, and Appreciable Numbers.- 5.2 Arithmetic of Hyperreals.- 5.3 On the Use of “Finite” and “Infinite”.- 5.4 Halos, Galaxies, and Real Comparisons.- 5.5 Exercises on Halos and Galaxies.- 5.6 Shadows.- 5.7 Exercises on Infinite Closeness.- 5.8 Shadows and Completeness.- 5.9 Exercise on Dedekind Completeness.- 5.10 The Hypernaturals.- 5.11 Exercises on Hyperintegers and Primes.- 5.12 On the Existence of Infinitely Many Primes.- II Basic Analysis.- 6 Convergence of Sequences and Series.- 6.1 Convergence.- 6.2 Monotone Convergence.- 6.3 Limits.- 6.4 Boundedness and Divergence.- 6.5 Cauchy Sequences.- 6.6 Cluster Points.- 6.7 Exercises on Limits and Cluster Points.- 6.8 Limits Superior and Inferior.- 6.9 Exercises on lim sup and lim inf.- 6.10 Series.- 6.11 Exercises on Convergence of Series.- 7 Continuous Functions.- 7.1 Cauchy’s Account of Continuity.- 7.2 Continuity of the Sine Function.- 7.3 Limits of Functions.- 7.4 Exercises on Limits.- 7.5 The Intermediate Value Theorem.- 7.6 The Extreme Value Theorem.- 7.7 Uniform Continuity.- 7.8 Exercises on Uniform Continuity.- 7.9 Contraction Mappings and Fixed Points.- 7.10 A First Look at Permanence.- 7.11 Exercises on Permanence of Functions.- 7.12 Sequences of Functions.- 7.13 Continuity of a Uniform Limit.- 7.14 Continuity in the Extended Hypersequence.- 7.15 Was Cauchy Right?.- 8 Differentiation.- 8.1 The Derivative.- 8.2 Increments and Differentials.- 8.3 Rules for Derivatives.- 8.4 Chain Rule.- 8.5 Critical Point Theorem.- 8.6 Inverse Function Theorem.- 8.7 Partial Derivatives.- 8.8 Exercises on Partial Derivatives.- 8.9 Taylor Series.- 8.10 Incremental Approximation by Taylor’s Formula.- 8.11 Extending the Incremental Equation.- 8.12 Exercises on Increments and Derivatives.- 9 The Riemann Integral.- 9.1 Riemann Sums.- 9.2 The Integral as the Shadow of Riemann Sums.- 9.3 Standard Properties of the Integral.- 9.4 Differentiating the Area Function.- 9.5 Exercise on Average Function Values.- 10 Topology of the Reals.- 10.1 Interior, Closure, and Limit Points.- 10.2 Open and Closed Sets.- 10.3 Compactness.- 10.4 Compactness and (Uniform) Continuity.- 10.5 Topologies on the Hyperreals.- III Internal and External Entities.- 11 Internal and External Sets.- 11.1 Internal Sets.- 11.2 Algebra of Internal Sets.- 11.3 Internal Least Number Principle and Induction.- 11.4 The Overflow Principle.- 11.5 Internal Order-Completeness.- 11.6 External Sets.- 11.7 Defining Internal Sets.- 11.8 The Underflow Principle.- 11.9 Internal Sets and Permanence.- 11.10 Saturation of Internal Sets.- 11.11 Saturation Creates Nonstandard Entities.- 11.12 The Size of an Internal Set.- 11.13 Closure of the Shadow of an Internal Set.- 11.14 Interval Topology and Hyper-Open Sets.- 12 Internal Functions and Hyperfinite Sets.- 12.1 Internal Functions.- 12.2 Exercises on Properties of Internal Functions.- 12.3 Hyperfinite Sets.- 12.4 Exercises on Hyperfiniteness.- 12.5 Counting a Hyperfinite Set.- 12.6 Hyperfinite Pigeonhole Principle.- 12.7 Integrals as Hyperfinite Sums.- IV Nonstandard Frameworks.- 13 Universes and Frameworks.- 13.1 What Do We Need in the Mathematical World?.- 13.2 Pairs Are Enough.- 13.3 Actually, Sets Are Enough.- 13.4 Strong Transitivity.- 13.5 Universes.- 13.6 Superstructures.- 13.7 The Language of a Universe.- 13.8 Nonstandard Frameworks.- 13.9 Standard Entities.- 13.10 Internal Entities.- 13.11 Closure Properties of Internal Sets.- 13.12 Transformed Power Sets.- 13.13 Exercises on Internal Sets and Functions.- 13.14 External Images Are External.- 13.15 Internal Set Definition Principle.- 13.16 Internal Function Definition Principle.- 13.17 Hyperfiniteness.- 13.18 Exercises on Hyperfinite Sets and Sizes.- 13.19 Hyperfinite Summation.- 13.20 Exercises on Hyperfinite Sums.- 14 The Existence of Nonstandard Entities.- 14.1 Enlargements.- 14.2 Concurrence and Hyperfinite Approximation.- 14.3 Enlargements as Ultrapowers.- 14.4 Exercises on the Ultrapower Construction.- 15 Permanence, Comprehensiveness, Saturation.- 15.1 Permanence Principles.- 15.2 Robinson’s Sequential Lemma.- 15.3 Uniformly Converging Sequences of Functions.- 15.4 Comprehensiveness.- 15.5 Saturation.- V Applications.- 16 Loeb Measure.- 16.1 Rings and Algebras.- 16.2 Measures.- 16.3 Outer Measures.- 16.4 Lebesgue Measure.- 16.5 Loeb Measures.- 16.6 ?-Approximability.- 16.7 Loeb Measure as Approximability.- 16.8 Lebesgue Measure via Loeb Measure.- 17 Ramsey Theory.- 17.1 Colourings and Monochromatic Sets.- 17.2 A Nonstandard Approach.- 17.3 Proving Ramsey’s Theorem.- 17.4 The Finite Ramsey Theorem.- 17.5 The Paris-Harrington Version.- 17.6 Reference.- 18 Completion by Enlargement.- 18.1 Completing the Rationals.- 18.2 Metric Space Completion.- 18.3 Nonstandard Hulls.- 18.4 p-adic Integers.- 18.5 p-adic Numbers.- 18.6 Power Series.- 18.7 Hyperfinite Expansions in Base p.- 18.8 Exercises.- 19 Hyperfinite Approximation.- 19.1 Colourings and Graphs.- 19.2 Boolean Algebras.- 19.3 Atomic Algebras.- 19.4 Hyperfinite Approximating Algebras.- 19.5 Exercises on Generation of Algebras.- 19.6 Connecting with the Stone Representation.- 19.7 Exercises on Filters and Lattices.- 19.8 Hyperfinite-Dimensional Vector Spaces.- 19.9 Exercises on (Hyper) Real Subspaces.- 19.10 The Hahn-Banach Theorem.- 19.11 Exercises on (Hyper) Linear Functionals.- 20 Books on Nonstandard Analysis.