Nonstandard Analysis by Jaap Ponstein
Nonstandard Analysis - Table of Contents
- Prologue & Preface
- 1. Generalities: Infinitesimals and History
- 1.1 Infinitesimals and other nonstandard numbers: getting acquainted
- 1.2 Other *-transforms; generating new numbers
- 1.3 Bound and free variables; prenex normal form
- 1.4 The purpose of nonstandard analysis
- 1.5 More about the *-transform; transfer
- 1.6 Standard, internal, and external constants
- 1.7 Infinitesimals in Greek geometry?
- 1.8 Infinitesimals in the 17th to 19th century
- 1.9 Infinitesimals in the 20th century
- 1.10 Introducing infinitesimals by filters
- 1.11 Basic assumptions of formalism
- 1.12 Basic assumptions of constructivism
- 1.13 Selecting basic assumptions naively
- 1.14 Basic definitions
- 1.15 Filters
- 1.16 Nature of free ultrafilters
- 2. Basic theory
- 2.1 Reviewing the introduction of Z, Q, and R
- 2.2 Introducing internal constants and equality
- 2.3 Identification of internal constants
- 2.4 Standard constants and basic results
- 2.5 External constants
- 2.6 The *-transform of operations and expressions
- 2.7 Los’ theorem and internal definition principle
- 2.8 Transfer and standard definition principle
- 2.9 The *-transform of attributes
- 2.10 *-IN, *-Z, *-Q, *-R definitions and properties
- 2.11 Overflow and underflow
- 2.12 *-IN and *-Z additional properties
- 2.13 *-Q, *-R properties and standard part
- 2.14 Alternative to introducing *-Z, *-Q, *-R
- 2.15 Summary of generating sequences
- 3. Some applications
- 3.1 Least upper bound theorem
- 3.2 Simplifying elementary calculus definitions
- 3.3 Continuity and limits for internal functions
- 3.4 Nonstandard characterizations of classical notions
- 3.5 Inverse functions
- 3.6 Differentiation
- 3.7 Integration
- 3.8 Pitfalls in nonstandard analysis
- 4. Some special topics
- 4.1 Principles of permanence
- 4.2 The saturation principle
- 4.3 Stirling’s formula
- 4.4 Nonstandard mathematics without choice
What You Will Learn in Nonstandard Analysis
Nonstandard Analysis by Jaap Ponstein is an authoritative, highly structured mathematical text that presents Abraham Robinson's nonstandard analysis using a accessible pedagogical approach. The book provides a mathematically sound framework for infinitesimals and hyperreal numbers, allowing traditional calculus and analysis concepts to be rephrased with intuitive rigor. It covers core topics such as Hyperreal Field Construction, Ultraproducts and Filter Foundations, Continuity and Differentiation via Infinitesimals, Nonstandard Integration Theory, and Applications to Topology and Functional Analysis.
This book serves as an invaluable guide for graduate students, logicians, and analysis researchers who want to explore alternative foundational methods in modern analysis. Ponstein carefully bridges abstract model theory and concrete mathematical applications, enabling readers to utilize infinitesimal techniques across classical real and complex analysis.
Due to its clear exposition and methodical development of nonstandard models, it is widely recognized as an outstanding resource for mathematical logic, real analysis, and foundation studies. Rather than treating nonstandard analysis as a purely logical curiosity, it demonstrates the practical elegance of infinitesimal methods in solving advanced mathematical problems.
Book Details & Specifications
Title:
Nonstandard Analysis by Jaap Ponstein
Publisher:
University of Groningen
Year:
2001
Pages:
465
Type:
PDF
Language:
English
ISBN-10 #:
B0006BNKKA
ISBN-13 #:
N\A
License:
University Educational Resource
Amazon:
Amazon
About the Author: Jaap Ponstein
The author Jaap Ponstein
was a distinguished Dutch mathematician and professor at the University of Groningen, known for his significant contributions to mathematical optimization, operations research, and real analysis.
Through Nonstandard Analysis, Ponstein demystified the logical machinery of hyperreal structures, rendering nonstandard techniques accessible to a broader mathematical audience. His clear writing style and logical progression make this text a masterpiece introduction to infinitesimal analysis and nonstandard methods.
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