In this post, we will see the book Stories About Sets N. Ya. Vilenkin.

# About the book

I want to tell the reader about the theory of sets in the same way, in which I learned it, by following the “corridor” course of study. Thus, our attention will be focused mainly on giving clear presentations of problems, discussing unexpected or surprising examples, quite often giving contradictory “naive” discussions. We shall find that the theory of functions of a real variable is richly endowed with all these. And if, after he has read this book, a high-school or college student wants to study the theory of sets or the theory of functions of a real variable more deeply, the author will feel that his book has been a success.

Professor Vilenkin has produced a small masterpiece that can be read with profit and delight by students of mathematics and laymen with an interest in mathematics. Slightly more than half the book explores the notion of cardinality of sets and the remainder traces the evolution of some of the most important concepts of m athem atics such as function, curve, surface and dimension.

The exposition combines informality with integrity of presentation and there is a wealth of unusual examples illustrating the paradoxical properties of curves and surfaces. Professor Vilenkin’s essay provides a royal road to the important concepts with which it is concerned.

…

Professor Vilenkin has produced a small masterpiece

which can be read with profit and delight by anybody,

beginning with high school juniors and seniors. Slightly

more than half of the book explores the notion of cardinality

of sets and the remainder traces the evolution of some of

the most important concepts of mathematics such as func-

tion, curve, surface, and dimension. The exposition combines

informality with integrity of presentation and there is a

wealth of unusual examples illustrating the paradoxical

properties of curves and surfaces. It is safe to say that

Professor Vilenkin’s essay provides a royal road to the

important concepts with which it is concerned.

The book was translated from Russian by Scripta Technica was published in 1968.

Credits to original uploader.

You can get the book here.

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# Contents

Foreword v

Preface vii

## 1. Some Extraordinary Properties of Infinite Sets

The Extraordinary Hotel, or the Thousand and First Journey of Ion the Quiet 4

From the Author 14

## 2. Sets and Operations on Sets

What Do We Mean by a Set? 16

How We Specify a Set 18

To Shave or Not to Shave? 21

The Empty Set 24

The Theory of Sets and Elementary Mathematics 26

Subsets 27

The Universal Set 29

The Intersection of Sets 29

Union of Sets 31

Partitioning of Sets 35

Subtraction of Sets 37

The Algebra of Sets 39

Boolean Algebras 41

## 3. The Cardinality of Sets

Equality between Sets 43

On the Dance Floor 44

For Every Flow There Is an Ebb 46

Can a Part Be Equal to the Whole? 47

Countable Sets 49

Algebraic Numbers 51

Unequal Sets 53

The Countable Set—The Smallest of the Infinite Sets 56

Uncountable Sets 57

The Census That Never Took Place 58

The Uncountability of the Continuum 61

The Existence of Transcendental Numbers 63

Long and Short Line Segments Have Equally Many Points 64

Segment and Square 66

Somehow One Problem Does Not Work Out 69

Is There a Set of Largest Cardinality ? 71

The Arithmetic of the Infinite 73

Infinite Exponents 76

On the Ordering of Numbers 78

Completely Ordered Sets 80

The Enigmatic Axiom 82

Two Apples from One 84

## 4. Remarkable Functions and Curves, or a Stroll through a Mathematical Art Museum

How the Notion of Function Developed 86

The Genie Escapes from the Bottle 91

Wet Points 93

The Devil’s Staircase 97

A Prickly Curve 99

A Closed Curve of Infinite Length 104

A Mathematical Carpet 107

Euclid Does Not Rely on Euclid 111

Are Rigorous Definitions Needed ? 112

A Curve Is the Path of a Moving Point 114

The Theorem Is Obvious, but the Proof Is Not 118

A Curve Passing through All the Points of a Square 120

Everything Had Come Unstrung 122

How to Make a Statue 124

Continua 126

Cantor Curves 128

Can the Area of a Curve Be Different from Zero ? 129

Domains without Area 133

Some Surprising Examples 135

Domains and Boundaries 137

The Great Irrigation Project 139

A “Nondissertable’’ Subject 141

The Inductive Definition of Dimension 144

The Article Is to Be Printed, Not Reviewed! 146

Conclusion 149

Exercises and Examples 150