Lectures on the Hyperreals: An Introduction to Nonstandard Analysis

ISBN-13: 9780387984643 / Angielski / Twarda / 1998 / 293 str.

Robert Goldblatt; R. Goldblatt

Opis

There are good reasons to believe that nonstandard analysis, in some ver- sion or other, will be the analysis of the future. KURT GODEL This book is a compilation and development of lecture notes written for a course on nonstandard analysis that I have now taught several times. Students taking the course have typically received previous introductions to standard real analysis and abstract algebra, but few have studied formal logic. Most of the notes have been used several times in class and revised in the light of that experience. The earlier chapters could be used as the basis of a course at the upper undergraduate level, but the work as a whole, including the later applications, may be more suited to a beginning graduate course. This prefacedescribes my motivationsand objectives in writingthe book. For the most part, these remarks are addressed to the potential instructor. Mathematical understanding develops by a mysterious interplay between intuitive insight and symbolic manipulation. Nonstandard analysis requires an enhanced sensitivity to the particular symbolic form that is used to ex- press our intuitions, and so the subject poses some unique and challenging pedagogical issues. The most fundamental ofthese is how to turn the trans- fer principle into a working tool of mathematical practice. I have found it vi Preface unproductive to try to give a proof of this principle by introducing the formal Tarskian semantics for first-order languages and working through the proofofLos's theorem.

Recenzje

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

Spis treści

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Are the Hyperreals?.- 1.1 Infinitely Small and Large.- 1.2 Historical Background.- 1.3 What Is a Real Number?.- 1.4 Historical References.- 2 Large Sets.- 2.1 Infinitesimals as Variable Quantities.- 2.2 Largeness.- 2.3 Filters.- 2.4 Examples of Filters.- 2.5 Facts About Filters.- 2.6 Zorn’s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