Arithmetic of infinity: From the winner of the 2010 Pythagoras International Prize in Mathematics

By Yaroslav D. Sergeyev

The first chapter of the book gives a brief description of the modern viewpoint on real numbers and presents the famous results of Georg Cantor regarding infinity.
The second chapter has a preparative character and links the first and the third parts of the book. On the one hand, it shows that the commonly accepted point of view on numbers and infinity is not so clear as it seems at first sight (for example, it leads to numerous paradoxes). On the other hand, the chapter contains preliminary observations that will be used in the constructive introduction of a new arithmetic of infinity, given in the third chapter.

This last part of the book contains the main results. It introduces notions of infinite and infinitesimal numbers, extended natural and real numbers, and operations with them. Surprisingly, the introduced arithmetical operations result in being very simple and are obtained as immediate extensions of the usual addition, multiplication, and division of finite numbers to infinite ones.
This simplicity is a consequence of a newly developed positional numeral system used to express infinite numbers. Finally, the chapter contains solutions to a number of paradoxes regarding infinity (we can say that the new approach allows us to avoid paradoxes) and some examples of applications.

In order to broaden the audience, the book was written as a popular one. The interested reader can find a number of technical articles of several researches that use the approach introduced here for solving a variety of research problems at the web page of the author.

The author Yaroslav D. Sergeyev is Distinguished Professor and Head of Numerical Calculus Laboratory at the University of Calabria, Italy. He is also Professor (part-time contract) at Lobachevsky Nizhni Novgorod State University, Russia. His research interests include numerical analysis, global optimization, infinity computing, set theory, number theory, fractals, and parallel computing. He has been awarded several national and international prizes (Pythagoras International Prize in Mathematics, Italy; Lagrange Lecture, Turin University, Italy; MAIK Prize for the best scientific monograph published in Russian, Moscow, etc.). His list of scientific publications contains more than 200 items. He is a member of editorial boards of 5 international journals and has given more than 50 plenary and keynote lectures at prestigious international congresses.

• Language: English
• Publisher: Yaroslav D. Sergeyev
• Release date: May 13, 2016
• ISBN: 9788889064016

Book preview

Preface

There is a concept which corrupts and upsets all others. I refer not to Evil, whose limited realm is that of ethics; I refer to the infinite.

Jorge Luis Borges,        

The Avatars of the Tortoise

Arithmetic is a branch of mathematics which studies finite numbers (natural, integers, rational, and real) and elementary properties of traditional operations (addition, division, multiplication, and subtraction) with them. It is normally learned at elementary school and then most people use calculators to perform arithmetical computations.

In contrast, studies of infinite and infinitesimal numbers are among the most difficult advanced areas of mathematics. In different periods of human history, mathematicians and physicists in order to solve theoretical and applied problems existing in their times developed mathematical languages that use different approaches to the ideas of infinity and infinitesimals (see works of Cantor, Conway, Godel, Hardy, Hilbert, Leibniz, Levi-Civita, Newton, Robinson, and Wallis in [1, 3, 6, 7, 8, 12, 13, 17, 19, 43] and references given therein). To emphasize importance of the subject it is sufficient to mention that the Continuum Hypothesis related to infinity has been included by David Hilbert as the Problem Number One in his famous list of 23 unsolved mathematical problems (see [8]) that have influenced strongly development of Mathematics in the 20th century.

Many approaches describing manipulations with infinities and infinitesimals are rather old: ancient Greeks following Aristotle distinguished the potential infinity from the actual infinity; John Wallis (see [43]) credited as the person who has introduced the infinity symbol ∞ (sometimes called the lemniscate, the name coming from the Latin word lemniscatus meaning ‘decorated with ribbons’) has published his work Arithmetica infinitorum in 1655; the foundations of analysis we use nowadays have been developed more than 200 years ago with the goal to develop mathematical tools allowing one to solve problems that were emerging in the world at that remote time; Georg Cantor (see [1]) has introduced his cardinals and ordinals more than 100 years ago, as well.

As a result, mathematical languages that we use now to work with infinities and infinitesimals do not reflect achievements made by Physics of the 20th century. Even the non-standard analysis created by Robinson (see [19]) in the middle of the 20th century has been also directed to a reformulation of the classical analysis (i.e., analysis created two hundred years before Robinson) in terms of infinitesimals and not to the creation of a new kind of analysis that would incorporate new achievements of Physics. In fact, Robinson wrote in paragraph 1.1 of his famous book [19]: ‘It is shown in this book that Leibniz’s ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical analysis and to many other branches of mathematics’ (the words classical analysis have been emphasized by the author of this book).

The goal of this small volume is to present a new type of arithmetic that would be closer to our modern views on the physical world and would allow us to treat infinite numbers in the same manner as we used to do with finite ones. The book shows that the problem of infinity can be considered in a different (with respect to the commonly accepted theory of infinity founded by Georg Cantor) coherent way allowing us to execute arithmetical operations with infinite numbers and to give detailed answers to many questions and paradoxes regarding infinite and infinitesimal quantities. Similarly to the introduction of negative numbers that has simplified solutions to many mathematical problems, construction of the new arithmetic simplifies solutions to problems dealing with infinity. It is important to clarify immediately that the new approach does not contradict Cantor, it just looks at infinities and infinitesimals in another way and with a different accuracy. Analogously, when we observe a physical object by eye and through a microscope we see different things with different accuracies.

Even though the book is mainly addressed to mathematicians, physicists, computer scientists, and students, the author tried to write it in such a way that any person having a high school education and who is interested in the foundations of these sciences should be able to understand it easily.

A few words about the history of the writing of this book. The present electronic edition is a revision and (the author hopes) an improvement of the paper edition published in 2003 (see [21]). During the ten years that have passed since its appearance the ideas presented in 2003 have been developed in many directions. The author has got several national and international research prizes and was invited to deliver numerous plenary and keynote lectures at prestigious scientific congresses. Two international conferences have been dedicated to the new approach to infinities and infinitesimals. Nowadays there exists an international scientific community working in this direction and publishing numerous research articles. The author invites the reader to follow new developments in this field at the internet page WWW.THEINFINITYCOMPUTER.COM

Introduction

Civilization advances by extending the number of important operations which we can perform without thinking about them.

Alfred North Whitehead,

An Introduction to Mathematics

There are some events and processes in our every day life that happen so frequently that we start to forget about their importance. For example, the sun rises every day and we do not take this into consideration. We do not think about the nature of this event, about its extreme importance for our existence. We just accept this event as a given and live without any undue concern about it.

There are many things we utilize in our every day life that are very useful and pleasant, for example: cellular phones, automobiles, water-pipe, etc. Usually, we do not think why and how these things work. We just use them as instruments to solve our problems.

Similarly to such events and instruments, there are ideas which we use without taking into consideration their nature and importance. We have studied them in our childhood and afterwards we have had no time nor necessity to think about them again. We return to reflect about them only if our life presses us to do so. Naturally, it is possible to give numerous examples (often contradictory) of such ideas that deal with various spheres of our life: (i) God exists; (ii) God does not exist; (iii) It is necessary to eat three (two, one, or four) times per day; (iv) Numbers can be summed up, etc.

Besides these useful and pleasant things, there is a set of events, objects, and ideas about which we prefer not to think because they are difficult or unpleasant. The idea of infinity belongs to this group. Moreover, infinity is both difficult and unpleasant. It is difficult because the human mind is not able to imagine infinity (just try to think a little bit about it to understand the truth of this phrase). As the result, infinity becomes unpleasant because we do not enjoy the feeling of our impotence.

This small book deals with a pleasant concept of finite numbers and a very unpleasant concept of infinity. People start to study both at school, then possibly they reconsider the subject at colleges or universities. Later they just use finite numbers and do not think about infinity.

However, these two subjects are among the most fundamental and have attracted the attention of the most brilliant thinkers throughout the whole history of humanity. Arabic, Indian, and Babylonian mathematicians worked hard on these problems. Aristotle, Archimedes, Euclid, Eudoxus, Parmenides, Plato, Pythagoras, and Zeno dealt with these problems in ancient Greek times. In the years 1500–1900 important contributions were made by such eminent researchers as Bolzano, Briggs, Cantor, Cauchy, Dedekind, Descartes, Dirichlet, Euler, Hermite, Leibniz, Lindemann, Liouville, Napier, Newton, Mercator, Peano, Stevin, Wallis, and Weierstrass. In the 20th century new exciting results have been obtained by Brouwer, Cohen, Frege, Gelfond, Gödel, Hilbert, Robinson, Scott, and Solovay.

Introduction of the ideas of the number line, positional number systems, negative numbers, zero, rational and irrational numbers, limits, continuum hypothesis, problems of consistency and completeness are among the major milestones of these impressive research efforts.

In this book, we shall not consider historical, philosophical, and religious aspects of problems arising when you consider infinity and numeral systems (nowadays anyone interested in these topics can easily retrieve the whole necessary information from the Internet). We start directly with a brief description of the modern viewpoint on real numbers and present the famous results of Georg Cantor¹ regarding infinity. These topics form the first chapter of the book.

The second chapter has a preparative character and links the first and the third parts of the book. On the one hand, it shows that the