Cambridge, 1906

Preface

The present volume is an attempt at a simple presentation of one of the most recent branches of mathematical science. It ha involved an amount of labour which would seem to the average reader quite out of proportion to the size of the book. yet I can scarcely hope that the mode of presentation will appeal equally to all mathematicians. There are no definitely accepted landmarks in the didactic treatment of Georg Cantor's magnificent theory, which is the subject of the present volume. A few of the most modern books on the Theory of Functions devote some pages to the establishment of certain results belonging to our subject, and required for the special purposes in hand. There is moreover in existence the first half of Schoenflies's useful *Bericht über die Mengenlehre.* The philosophical point of view is discussed to some extent in Russell's *Principles of Mathematics*. But we may fairly claim that the present work is the first attempt at a systematic exposition of the subject as a whole.

....

On the other hand, imperfect though the book is felt to be, it is hoped that it may prove of use to a somewhat large class of readers. As far as the professional mathematician is concerned, it may be confidently asserted that a grasp of the Theory of Sets of Points is indispensable. Wherever he has to deal—and where does he not?—with an infinite number of operations, he is treading on ground full of pitfalls, one or more of which may well prove fatal to him, if he is unprovided by the clue to furnish which is the object of the present volume.

In subjects as wide apart as Projective Geometry, Theory of Functions of a Complex Variable, the Expansions of Astronomy, Calculus of Variations, Differential Equations, mistakes have in fact been made by mathematicians of standing, which even a slender grasp of the Theory of Sets of Points would have enabled them to avoid. It can scarcely be doubted that the near future will see a marked influence exerted by our theory on the language and conceptions of Applied Mathematics and Physics. To the philosophical reader on the other hand and to the general public with mathematical interests the subject presents the advantage, as compared with other of the more recent developments of mathematical science, that it is less technical and requires a smaller mathematical equipment than most of them.

...Any reference to the constant assistance which I have received during my work from my wife is superfluous, since, with the consent of the Syndics of the Press, her name has been associated with mine on the title-page.

W. H. Young

Heswall. May, 1906.

**Table of Contents**

- Rational and Irrational Numbers
- Introductory
- Sets and sequences
- Irrational numbers
- Magnitude and equality
- The number ∞
- Limit
- Algebraic and transcendental numbers

- Representation of Numbers on the Straight Line
- The projective scale
- Interval between two numbers

- The Descriptive Theory of Linear Sets of Points
- Sets of points. Sequences. Limiting points
- Fundamental sets
- Derived sets. Limiting points of various orders
- Deduction
- Theorems about a set and its derived and deduced sets
- Intervals and their limits
- Upper and lower limit

- Potency, and the Generalised Idea of a Cardinal Number
- Measurement and potencies
- Countable sets
- Preliminary definitions of addition and multiplication
- Countable sets of intervals
- Some theorems about countable sets of points
- More than countable sets
- The potency
*c* - Symbolic equations involving the potency
*c* - Limiting points of countable and more than countable degree
- Closed and perfect sets
- Derived and deduced sets
- Adherences and coherences
- The ultimate coherence
- Tree illustrating the theory of adherences and coherences
- Ordinary inner limiting sets
- Relation of any set to the inner limiting set of a series of sets of intervals containing the given set
- Sets of the first and second category
- Generality of the class of inner and outer limiting sets

- Content
- Meaning of content
- Content of a finite number of non-overlapping intervals
- Extension to an infinite number of non-overlapping intervals
- Definition of content of such a set of intervals
- Examples of such sets of intervals
- Content of such a set and potency of complementary set of points
- Properties of the content of such a set of intervals
- Addition Theorem for the content of sets of intervals
- Content of a closed set of points
- Addition Theorem for the content of closed sets of points
- Connexion between the content and the potency
- Historical note on the theory of content
- Content of any closed component of an ordinary inner limiting set
- (Content of any closed component of an ordinary inner limiting set, continued)
- Content of any closed component of a generalised inner limiting set, defined by means of closed sets
- Open sets
- The (inner) content
- The (inner) addition Theorem
- Possible extension of the (inner) addition Theorem
- The (inner) additive class, and the addition theorem for the (inner) contents
- Reduction of the classification of open sets to that of sets of zero (inner) content
- The (outer) content
- Measurable sets
- An ordinary inner or outer limiting set is measurable
- The (inner) additive class consists of measurable sets
- The (outer) additive class consists of measurable sets
- Outer and inner limiting sets of measurable sets
- Theorem for the (outer) content analogous to Theorem 20 of § 52
- Connexion of the (outer) content with the theory of adherences and coherences
- The (outer) additive class
- The additive class
- Content of the irrational numbers

- Order
- Order is a property of the set
*per se* - The characteristic of order
- Finite ordinal types
- Order of the natural numbers
- Orders of closed sequences, etc.
- Graphical and numerical representation
- The rational numbers. Close order
- Condition that a set in close order should be dense everywhere
- Limiting points of a set in close order
- Ordinally closed, dense in itself, perfect. Ordinal limiting point
- Order of the continuum
- Order of the derived and deduced sets
- Well-ordered sets
- Multiple order

- Order is a property of the set
- Cantor's Numbers
- Cardinal numbers
- General definition of the word "set"
- The Cantor-Bernstein-Schroeder Theorem
- Greater, equal and less
- The addition and multiplication of potencies
- The Alephs
- Transfinite ordinals of the second class
- Ordinals of higher classes
- The series of Alephs
- The theory of ordinal addition
- The law of ordinal multiplication

- Preliminary Notions of Plane Sets
- Space of any countable number of dimensions as fundamental region
- The two-fold continuum
- Dimensions of the fundamental region
- Cantor's (1,1)-correspondence between the points of the plane, or
*n*-dimensional space and those of the straight line - Analogous correspondence when the space is of a countably infinite number of dimensions
- Continuous representation
- Peano's continuous representation of the points of the unit square on those of the unit segment
- Discussion of the term "space-filling curve"
- Moore's crinkly curves
- Continuous (1,1)-correspondence between the points of the whole plane and those of the interior of a circle of radius unity
- Definition of a plane set of points
- Limiting points, isolated points, sequences, etc. Examples of plane perfect sets
- Plane sequence in any set corresponding to any limiting point
- The minimum distance between two sets of points

- Regions and Sets of Regions
- Plane elements
- Primitive triangles
- Definitions of a domain, a region, etc.
- Internal and external points of a region. Boundary and edge points
- Ordinary external points and external boundary points
- Describing a region
- Two internal points of a region can be joined by a finite set of generating triangles
- The Chow
- The rim
- Sections of a region
- The span
- Discs
- Case when the inner limiting set of a series of regions is a point, or a stretch
- Weierstrass's Theorem
- General discussion of the inner limiting set of a series of regions
- (General discussion of the inner limiting set of a series of regions, continued)
- Finite and infinite regions
- The domain as space element
- The rim is a perfect set dense nowhere
- Sets of regions
- Classification of the points of the plane with reference to a set of regions
- Cantor's Theorem of non-overlapping regions. The extended Heine-Borel Theorem, etc.
- The black regions of a closed set
- Connected sets
- The inner limiting set of a series of regions, if dense nowhere, is a curve
- Simple polygonal regions
- The outer rim
- General form of a region
- The black region of a closed set containing no curves
- A continuous (1,1)-correspondence between the points of a region of the plane and a segment of the straight line is impossible
- Uniform continuity

- Curves
- Definition and fundamental properties of a curve
- Branches, end-points and closed curves
- Jordan curves
- Sets of arcs and closed sets of points on a Jordan curve

- Potency of Plane Sets
- The only potencies in space of a countable number of dimensions are those which occur on the straight line
- Countable sets
- The potency
*c* - Limiting points of countable and more than countable degree
- Ordinary inner limiting sets
- Relation of any set to the inner limiting set of a series of sets of regions containing the given set

- Plane Content and Area
- Various kinds of content which occur in space of more than one dimension
- The theory of plane content in the plane
- Content of triangles and regions
- Content of a closed set
- Area of a region
- A simply connected non-quadrable region, whose rim is a Jordan curve of positive content
- Connexion between the potency of a closed set and the content of its black regions
- Content of a countable closed set is zero
- Content of any closed component of an ordinary inner limiting set
- Measurable sets. Inner and outer measures of the content
- Calculation of the plane content of closed sets
- Upper and lower
*n*-tuple and*n*-fold integrals - Upper and lower semi-continuous functions
- The associated limiting functions of a function
- Calculation of the upper integral of an upper semi-continuous function
- Application of §§ 159–162 to the calculation of the content by integration
- Condition that a plane closed set should have zero content
- Expression for the content of a closed set as a generalised or Lebesgue integral
- Calculation of the content of any measurable set

- Length and Linear Content
- Length of a Jordan curve
- Calculation of the length of a Jordan curve
- Linear content on a rectifiable Jordan curve
- General notions on the subject of linear content
- Definition of linear content
- Alternative definition of linear content
- Linear content of a finite arc of a rectifiable Jordan curve
- Linear content of a set of arcs on a rectifiable Jordan curve
- Linear content of a countable closed set of points