# Lectures on the Hyperreals: An Introduction to Nonstandard Analysis

## Goldblatt, Robert; Axler, S.; Gehring, F. W.; Halmos, P. R.

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An introduction to nonstandard analysis based on a course given by the author. It is suitable for beginning graduates or upper undergraduates, or for self-study by anyone familiar with elementary real analysis. It presents nonstandard analysis not just as a theory about infinitely small and large numbers, but as a radically different way of viewing many standard mathematical concepts and constructions. It is a source of new ideas, objects and proofs, and a wealth of powerful new principles of reasoning. The book begins with the ultrapower construction of hyperreal number systems, and proceeds to develop one-variable calculus, analysis and topology from the nonstandard perspective. It then sets out the theory of enlargements of fragments of the mathematical universe, providing a foundation for the full-scale development of the nonstandard methodology. The final chapters apply this to a number of topics, including Loeb measure theory and its relation to Lebesgue measure on the real line. Highlights include an early introduction of the ideas of internal, external and hyperfinite sets, and a more axiomatic set-theoretic approach to enlargements than is usual.

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Recensione:

R. Goldblatt

Lectures on the Hyperreals

An Introduction to Nonstandard Analysis

"Suitable for a graduate course . . . could be covered in an advanced undergraduate course . . . The author’s ideas on how to achieve both intelligibility and rigor . . . will be useful reading for anyone intending to teach nonstandard analysis."—AMERICAN MATHEMATICAL SOCIETY

Contenuti:

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Are the Hyperreals?.- 1.1 Infinitely Small and Large.- 1.2 Historical Background.- 1.3 What Is a Real Number?.- 1.4 Historical References.- 2 Large Sets.- 2.1 Infinitesimals as Variable Quantities.- 2.2 Largeness.- 2.3 Filters.- 2.4 Examples of Filters.- 2.5 Facts About Filters.- 2.6 Zorn’s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.

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## 1.Lectures on the Hyperreals: An Introduction to Nonstandard Analysis

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