How to Measure the Infinite

Mathematics with Infinite and Infinitesimal Numbers

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.
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