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How to Measure the Infinite Mathematics with Infinite and Infinitesimal Numbers 1st edition by Vieri Benci, Mauro Di Nasso ISBN 9812836373 978-9812836373

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ISBN 10: 9812836373
ISBN 13: 978-9812836373
Author: Vieri Benci, Mauro Di Nasso

This book contains an original introduction to the use of infinitesimal and infinite numbers, namely, the Alpha-Theory, which can be considered as an alternative approach to nonstandard analysis.

The basic principles are presented in an elementary way by using the ordinary language of mathematics; this is to be contrasted with other presentations of nonstandard analysis where technical notions from logic are required since the beginning. Some applications are included and aimed at showing the power of the theory.

The book also provides a comprehensive exposition of the Theory of Numerosity, a new way of counting (countable) infinite sets that maintains the ancient Euclid’s Principle: “The whole is larger than its parts”. The book is organized into five parts: Alpha-Calculus, Alpha-Theory, Applications, Foundations, and Numerosity Theory.

Readership: Advanced undergraduate and graduate students in mathematics and philosophy.

How to Measure the Infinite Mathematics with Infinite and Infinitesimal Numbers 1st Table of contents:

Historical Introduction

1. Ancient times

2. The rise of calculus

3. The ban of infinitesimals

4. Non-Archimedean mathematics

Part 1. Alpha-Calculus

Chapter 1. Extending the Real Line

1. Ordered fields

2. Infinitesimal numbers

3. The smallest non-Archimedean field

4. Proper extensions of the real line

5. Standard parts

6. Monads and galaxies

Chapter 2. Alpha-Calculus

1. The axioms of Alpha-Calculus

2. First properties of Alpha-Calculus

3. Hyper-extensions of sets of reals

4. The Alpha-measure and the qualified sets

5. The transfer principle, informally

6. Hyper-extensions of functions

7. Some more relevant basic properties

8. Hyper-extensions of sets of numbers

9. The Qualified Set Axiom

10. Rings and ideals

11. Models of Alpha-Calculus

Chapter 3. Infinitesimal Analysis by Alpha-Calculus

1. The normal forms

2. Infimum and supremum

3. Continuity

4. Uniform continuity

5. Derivatives

6. Limits

7. Alpha-limit versus limit

8. The order of magnitude

9. Hyperfinitely long sums and series

10. The grid integral

11. Equivalences with the “standard” definitions

12. Grid integral versus Riemann integral

13. Remarks and comments

Part 2. Alpha-Theory

Chapter 4. Introducing the Alpha-Theory

1. The axioms of Alpha-Theory

2. First properties of Alpha-Theory

3. Some detailed proofs

4. Hyper-images of sets

5. Hyper-images of functions

6. Functions of several variables

7. Hyperfinite sets

8. Hyperfinite sums

9. Internal objects

10. Remarks and comments

Chapter 5. Logic and Alpha-Theory

1. Some logic formalism

2. Transfer principle

3. Transfer and internal sets

4. The transfer as a unifying principle

5. Remarks and comments

Chapter 6. Complements of Alpha-Theory

1. Overspill and underspill

2. Countable saturation

3. Cauchy infinitesimal principle

4. The S-topology

5. The topology of Alpha-limits

6. Superstructures

7. Models of Alpha-Theory

8. Existence of reflexive Alpha-morphisms

9. Remarks and comments

Part 3. Applications

Chapter 7. First Applications

1. The real line as a quotient of hyperrationals

2. Ramsey’s Theorem

3. Grid functions

4. Grid differential equations

5. Peano’s theorem

Chapter 8. Gauge Spaces

1. Main definitions

2. Gauge Abelian spaces

3. Topological notions for gauge spaces

4. Topology theorems in gauge spaces

5. Gauge spaces versus topological spaces

6. The Epsilon-gauges

Chapter 9. Gauge Quotients

1. Definition of gauge quotients

2. The differential Epsilon-ring

3. Distributions as a gauge quotient

4. Distributions as functionals

5. Grid functions and distributions

Chapter 10. Stochastic Differential Equations

1. Preliminary remarks on the white noise

2. Stochastic grid equations

3. Itô’s formula for grid functions

4. The Fokker-Plank equation

5. Remarks and comments

Part 4. Foundations

Chapter 11. Ultrafilters and Ultrapowers

1. Filters and ultrafilters

2. Ultrafilters as measures and as ideals

3. Ultrapowers, the basic examples

4. Ultrapowers as models of Alpha-Calculus

Chapter 12. The Uniqueness Problem

1. Isomorphic models of Alpha-Calculus

2. Equivalent models of Alpha-Calculus

3. Uniqueness up to countable equivalence

Chapter 13. Alpha-Theory and Nonstandard Analysis

1. Nonstandard analysis, a quick presentation

2. Alpha-Theory is more general than nonstandard analysis

3. Remarks and comments

Chapter 14. Alpha-Theory as a Nonstandard Set Theory

1. The axioms of AST

2. Alpha Set Theory versus ZFC

3. Cauchy infinitesimal principle and special ultrafilters

4. The strength of Cauchy infinitesimal principle

5. The strength of a Hausdorff S-topology

6. Remarks and comments

Part 5. Numerosity Theory

Chapter 15. Counting Systems

1. The idea of counting

2. Cardinals and ordinals as counting systems

3. Three different ways of counting

4. The equisize relation

Chapter 16. Alpha-Theory and Numerosity

1. Labelled sets

2. Alpha-numerosity

3. Finite parts and sets of functions

4. Point sets of natural numbers

5. Numerosity of sets of natural numbers

6. Properties of Alpha-numerosity

7. Numerosities of sets of rational numbers

8. Zermelo’s principle and Alpha-numerosity

9. Asymptotic density and numerosity

Chapter 17. A General Numerosity Theory for Labelled Sets

1. Definition and first properties

2. From numerosities to Alpha-Calculus

3. Remarks and comments

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