## Stories About Sets – Vilenkin

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

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.

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
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