How to Measure the Infinite

Mathematics with Infinite and Infinitesimal Numbers

Vieri Benci, Mauro Di Nasso

• Publisher: WSPC
• Topic: Mathematics
• eBook ISBN:
  9789813276604
• Subtopic: Logic in Mathematics
• Index: Mathematics

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.

Contents:

• Historical Introduction
• Alpha-Calculus:
  • Extending the Real Line
  • Alpha-Calculus
  • Infinitesimal Analysis by Alpha-Calculus
• Alpha-Theory:
  • Introducing the Alpha-Theory
  • Logic and Alpha-Theory
  • Complements of Alpha-Theory
• Applications:
  • First Applications
  • Gauge Spaces
  • Gauge Quotients
  • Stochastic Differential Equations
• Foundations:
  • Ultrafilters and Ultrapowers
  • The Uniqueness Problem
  • Alpha-Theory and Nonstandard Analysis
  • Alpha-Theory as a Nonstandard Set Theory
• Numerosity Theory:
  • Counting Systems
  • Alpha-Theory and Numerosity
  • A General Numerosity Theory for Labelled Sets

Readership: Advanced undergraduate and graduate students in mathematics and philosophy.Nonstandard Analysis;Calculus;Infinitesimal Numbers;Numerosities;?-Theory;Alpha-Theory;Nonstandard Models;Nonstandard Set Theories0 Key Features:

• Provides a comprehensive exposition of a new way of counting infinite sets
• Presents the α-theory in elementary terms by using only the most basic notions of mathematics

Part 1

Alpha-Calculus

The Alpha-Calculus is a new elementary method for developing a calculus grounded on the use of infinitesimal and infinite numbers. Basically, it is a simplified version of nonstandard analysis, that still allows for a complete and rigorous treatment of all the basics of calculus. Most notably, Alpha-Calculus is expressed in the everyday language of mathematics and only assumes a few basic notions from algebra.

In the first chapter, we study the class of the ordered fields that properly extend the real line. The Archimedean property necessarily fails, and one can distinguish between infinitely small, finite, and infinitely large numbers. A few basic algebraic properties of numbers according to their “size” is then established, and the fundamental notion of standard part is introduced.

After “warming-up” with the general properties of fields that extend the real line, in the second chapter we introduce the axioms of Alpha-Calculus that rule the properties of the hyperreal line R^∗. The hyperreal numbers are an ordered field that extends the real line, and that satisfies remarkable additional properties. Most notably, every subset A ⊆ R and every real function f : A → B can be canonically extended to a subset A^∗ ⊆ R^∗ (called the hyperextension of A), and to a function f^∗ : A^∗ → B^∗ (the hyperextension of f), respectively. This is done in such a way that all “elementary properties” are preserved under the hyper-extensions.¹⁴

In the third chapter, the main basic notions and results of calculus are developed within the Alpha-Theory, and a selection of relevant classic results – such as the Extreme Value and the Intermediate Value theorems, and the Fundamental Theorem of Calculus – are proved. In particular, a notion of grid integral is introduced, which generalizes the usual Riemann integral and makes sense for all real functions.

According to the pedagogical purposes of this part, in the exposition the transfer principle is never mentioned. Rather, on a case by case basis, we prove directly only those instances that are actually needed to obtain the desired results.

¹⁴ The notion of “elementary property” will be given a precise meaning in Chapter 5 by using the formalism of first-order logic.

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Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Preface
• Plan of the Book
• Notation
• Historical Introduction
• Part 1. Alpha-Calculus
• Part 2. Alpha-Theory
• Part 3. Applications
• Part 4. Foundations
• Part 5. Numerosity Theory
• Bibliography
• Index