How to Measure the Infinite

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 IntroductionAlpha-Calculus:Extending the Real LineAlpha-CalculusInfinitesimal Analysis by Alpha-CalculusAlpha-Theory:Introducing the Alpha-TheoryLogic and Alpha-TheoryComplements of Alpha-TheoryApplications:First ApplicationsGauge SpacesGauge QuotientsStochastic Differential EquationsFoundations:Ultrafilters and UltrapowersThe Uniqueness ProblemAlpha-Theory and Nonstandard AnalysisAlpha-Theory as a Nonstandard Set TheoryNumerosity Theory:Counting SystemsAlpha-Theory and NumerosityA General Numerosity Theory for Labelled Sets
Readership: Advanced undergraduate and graduate students in mathematics and philosophy.Nonstandard Analysis;Calculus;Infinitesimal Numbers;Numerosities;α-Theory;Alpha-Theory;Nonstandard Models;Nonstandard Set Theories0Key Features:Provides a comprehensive exposition of a new way of counting infinite setsPresents the α-theory in elementary terms by using only the most basic notions of mathematics