Introduction to Logic and to the Methodology of Deductive Sciences

• IV •

21. Classes and their elements

Apart from separate individual things, which we shall also, for short, call INDIVIDUALS, logic is concerned with CLASSES of things; in everyday life as well as in mathematics, classes are more often referred to as SETS. Arithmetic, for instance, frequently deals with sets of numbers, and in geometry our interest attaches itself not so much to single points as to point sets (namely, to geometrical configurations). Classes of individuals are called CLASSES OF THE FIRST ORDER. Comparatively more rarely we also meet in our investigations with CLASSES OF THE SECOND ORDER, that is, with classes which consist, not of individuals, but of classes of the first order. Sometimes even CLASSES OF THE THIRD, FOURTH, … ORDERS have to be dealt with. Here we shall be concerned almost exclusively with classes of the first order, and only exceptionally—as in Section 26—we shall have to deal with classes of the second order; our considerations can, however, be applied with practically no changes to classes of any order.

In order to differentiate between individuals and classes (and also between classes of different orders), we employ as variables letters of different shape and belonging to different alphabets. It is customary to designate individual things such as numbers, and classes of such things, by the small and capital letters of the English alphabet, respectively. In elementary geometry the opposite notation is the accepted one, capital letters designating points and small letters (of the English or Greek alphabets) designating point sets.

That part of logic in which the class concept and its general properties are examined is called the THEORY OF CLASSES; sometimes this theory is also treated as an independent mathematical discipline under the name of the GENERAL THEORY OF SETS.¹

Of fundamental character in the theory of classes are such phrases as:

the thing x is an element (or a member) of the class K,

the thing x belongs to the class K,

the class K contains the thing x as an element (or a member);

we consider these expressions as having the same meaning and, for the sake of brevity, replace them by the formula:

Thus, if I is the set of all integers, the numbers 1, 2, 3, … are its elements, whereas the numbers do not belong to the set; hence, the formulas:

are true, while the formulas:

are false.

22. Classes and sentential functions with one free variable

We consider a sentential function with one free variable, for instance:

x > 0

If we prefix the words:

(I) the set of all numbers x such that

to that function, we obtain the expression:

the set of all numbers x such that x > 0.

This expression designates a well-determined set, namely the set of all positive numbers; it is the set having as its elements those, and only those, numbers which satisfy the given function. If we denote this set by the symbol “P”, our function becomes equivalent to:

We may apply an analogous procedure to any other sentential function. In arithmetic, we can obtain in this way various sets of numbers, for instance the set of all negative numbers or the set of all numbers which are greater than 2 and less than 5 (that is which satisfy the function “x > 2 and x < 5”). This procedure plays also an important role in geometry, especially in defining new kinds of geometrical configurations; the surface of a sphere is defined, for instance, as the set of all points of the space which have a definite distance from a given point. It is customary in geometry to replace the words “the set of all points” by “the locus of the points.”

We will now put the above remarks in a general form. It is assumed in logic that, to every sentential function containing just one free variable, say “x”, there is exactly one corresponding class having as its elements those, and only those, things x which satisfy the given function. We obtain a designation for that class by putting in front of the sentential function the following phrase, which belongs to the fundamental expressions of the theory of classes:

(II) the class of all things x such that.

If we denote further the class in question by a simple symbol, say “C”, the formula:

will—for any x—be equivalent to the original sentential function.

Hence it is seen that any sentential function containing “x” as the only free variable can be transformed into an equivalent function of the form:

where in place of “K” we have a constant denoting a class; one may, therefore, consider the latter formula as the most general form of a sentential function with one free variable.

The phrases (I) and (II) are sometimes replaced by symbolic expressions; we can, for instance, agree to use the following symbol for this purpose:

*Let us now consider the following expression:

1 belongs to the set of all numbers x such that x > 0,

which can also be written in symbols only:

This expression is obviously a sentence, and even a true sentence; it expresses, in a more complicated form, the same thought as the simple formula:

1 > 0

Consequently, this expression cannot contain any free variable, and the variable “x” occurring in it must be a bound variable. Since, on the other hand, we do not find in the above expression any quantifiers, we arrive at the conclusion that such phrases as (I) or (II) function like quantifiers, that is, they bind variables, and must, therefore, be counted among the operators (cf. Section 4).

It should be added that we frequently prefix an operator like (I) or (II) to sentential functions which contain—besides “x”—other free variables (this occurs in nearly all cases in which such operators are applied in geometry). The expressions thus obtained, for instance:

the set of all numbers x such that x > y

do not designate, however, any definite class; they are designatory functions in the meaning established in Section 2, that is, they become designations of classes if we replace in them free variables (but not “x”) by suitable constants, for instance, “y” by “0” in the example just given.*

It is frequently said of a sentential function with one free variable that it expresses a certain property of things,—a property possessed by those, and only those, things which satisfy the sentential function (the sentential function “x is divisible by 2”, for example, expresses a certain property of the number x, namely, divisibility by 2, or the property of being even). The class corresponding to this function contains as its elements all things possessing the given property, and no others. In this manner it is possible to correlate a uniquely determined class with every property of things. And also, conversely, with every class there is correlated a property possessed exclusively by the elements of that class, namely, the property of belonging to that class. It is, accordingly, in the opinion of numerous logicians, unnecessary to distinguish at all between the two concepts of a class and of a property; in other words, a special “theory of properties” is dispensable,—the theory of classes being perfectly sufficient.

As an application of these remarks we shall give a new formulation of LEIBNIZ’s law. The original one (in Section 17) contained the term “property”; in the following, entirely equivalent, formulation we employ the term “class” instead:.

x = y if, and only if, every class which contains any one of the things
x and y as an element also contains the other as an element.

As can be seen from this formulation of LEIBNIZ’S law, it is possible to define the concept of identity in terms of the theory of classes.

23. Universal class and null class

As we already know, to any sentential function with one free variable there corresponds the class of all objects satisfying this function. This can now be applied to the following two particular functions:

The first of these functions is obviously satisfied by every individual (cf. Section 17). The corresponding class,

therefore, contains as elements all individuals; we call this class the UNIVERSAL CLASS and denote it by the symbol “∨” (or “1”). The second sentential function, on the other hand, is satisfied by no thing. Consequently, the class corresponding to it,

called the NULL CLASS or EMPTY CLASS and denoted by “∧” (or “0”), contains no elements. We may now replace the sentential functions (I) by equivalent functions of the form:

namely by:

the first of which is satisfied by any individual, and the second by none.

Instead of using the general logical concept of individual within a particular mathematical theory, it is sometimes more convenient to specify exactly what is considered an individual thing within the framework of this theory; the class of all those things will then be denoted again by “∨” and will be called the UNIVERSE OF DISCOURSE of the theory. In arithmetic, for instance, it is the class of all numbers which forms the universe of discourse.

*It should be emphasized that ∨ is the class of all individuals but not the class containing as elements all possible things, thus also classes of first order, second order, and so on. The question arises whether such a class of all possible things exists at all, and more generally, whether we may consider “inhomogeneous” classes not belonging to a particular order and containing as elements individuals as well as classes of various orders. This question is closely related to the most intricate problems of contemporary logic, namely, to the so-called ANTINOMY OF RUSSELL and the THEORY OF LOGICAL TYPES.² A discussion of this question would trespass beyond the intended limits of this book. We will only remark here that the need for considering “inhomogeneous” classes occurs hardly ever in the whole of mathematics (except for the general theory of sets), and even more rarely in other sciences.*

24. Fundamental relations among classes

Various relations may hold between two classes K and L. It may, for instance, occur that every element of the class K is at the same time an element of the class L in which case the set K is said to be a SUBCLASS OF THE CLASS L or to be INCLUDED IN THE CLASS L, or to HAVE THE RELATION OF INCLUSION TO THE CLASS L; and the class L is said to COMPREHEND THE CLASS K AS A SUBCLASS. This situation is expressed, briefly, by either of the formulas:

K ⊂ L or L ⊃ K.

By saying that K is subclass of L it is not intended to preclude the possibility of L also being a subclass of K. In other words, K and L may be subclasses of each other and thus have all their elements in common; in this case it follows from a law (given below) of the theory of classes that K and L are identical. If, however, the converse relation does not hold, that is, if every element of the class K is an element of the class L, but if not every element of the class L is an element of the class K, then the class K is said to be a PROPER SUBCLASS or a PART OF THE CLASS L, and L is said to COMPREHEND K AS A PROPER SUBCLASS or AS A PART. For example, the set of all integers is a proper subset of the set of all rational numbers; a line comprehends each of its segments as a part.

Two classes K and L are said to OVERLAP or to INTERSECT if they have at least one element in common and if, at the same time, each contains elements not contained in the other. If two classes have each at least one element (i.e. if they are not empty), but if they have no element in common, they are called MUTUALLY EXCLUSIVE or DISJOINT. A circle, for instance, intersects any straight line drawn through its center, but it is disjoint from any straight line whose distance from the center is greater than the radius. The set of all positive numbers and the set of all rational numbers overlap, but the set of positive and the set of negative numbers are mutually exclusive.

Let us give some examples of laws concerning the relations between classes mentioned above.

For any class K, K ⊂ K.

If K ⊂ L and L ⊂ K, then K = L.

If K ⊂ L and L ⊂ M, then K ⊂ M.

If K is a non-empty subclass of L, and if the classes L and M are disjoint, then the classes K and M are disjoint.

The first of these statements is called the LAW OF REFLEXIVITY for inclusion or the class-theoretical LAW OF IDENTITY. The third is known as the LAW OF TRANSITIVITY for inclusion; together with the fourth statement and others of a similar structure they form a group of statements which are called LAWS OF THE CATEGORICAL SYLLOGISM.

A characteristic property of the universal and null classes in connection with the concept of inclusion is expressed in the following law:

For any class K, ∨ ⊃ K and ∧ ⊂ K.

This statement, particularly in view of its second part referring to the null class, seems to many people somewhat paradoxical. In order to demonstrate this second part, let us consider the implication:

Whatever we substitute here for “x” (and “K”), the antecedent of the implication will be a false sentence, and hence the whole implication a true sentence (the implication—as the mathematicians sometimes say—is satisfied “vacuously”). We may, thus, say that whatever is an element of the class ∧ is also an element of the class K, and hence, by the definition of inclusion, that ∧ ⊂ K. —In an analogous way the first part of the law can be demonstrated.

It is easy to see that between any two classes one of the relations considered here has to hold; the following law is to this effect:

If K and L are two arbitrary classes, then either K = L or K is a proper subclass of L, or K comprehends L as a proper subclass, or K and L overlap, or finally K and L are disjoint; no two of these relations can hold simultaneously.

In order to get a clear intuitive understanding of this law it is best to think of the classes K and L as geometrical figures and to imagine all the possible positions in which these two figures may be with respect to each other.

The relations which have been dealt with in this section may be called the FUNDAMENTAL RELATIONS AMONG CLASSES.³

The whole of the old traditional logic (cf. Section 6) can almost entirely be reduced to the theory of the fundamental relations among classes, that is, to a small fragment of the entire theory of classes. Outwardly these two disciplines differ by the fact that, in the old logic, the concept of a class does not appear explicitly. Instead of saying, for instance, that the class of horses is contained in the class of mammals, one used to say in the old logic that the property of being a mammal belongs to all horses, or, simply, that every horse is a mammal. The most important laws of traditional logic are those of the categorical syllogism which correspond precisely to the laws of the theory of classes that we stated above and named after them. For example, the first of the laws of syllogism given above assumes the following form in the old logic:

If every M is P and every S is M, then every S is P.

This is the most famous of the laws of traditional logic, known as the law of the syllogism BARBARA.

25. Operations on classes

We shall now concern ourselves with certain operations which, if performed on given classes, yield new classes.

Given any two classes K and L, one can form a new class M which contains as its elements those, and only those, things which belong to at least one of the classes K and L; the class M, one might say, results from the class K by adjoining to it the elements of the class L. This operation is called ADDITION OF CLASSES, and the class M is referred to as the SUM or UNION OF THE CLASSES K AND L, designated by the symbol:

K ∪ L (or K + L).

Another operation on two classes K and L, called MULTIPLICATION OF CLASSES, consists in forming a new class M whose elements are those, and only those, things which belong to both K and L; this class M is called the PRODUCT or INTERSECTION OF THE CLASSES K AND L and is designated by the symbol:

K ∩ L (or K · L).

These two operations are frequently applied in geometry; sometimes it is very convenient to define with their help new kinds of geometrical figures. Suppose, for instance, we know already what is meant by a pair of supplementary angles; then the half-plane—that is, the straight angle—may be defined as the union of two supplementary angles (an angle here being considered as an angular region, that is, as a part of the plane, bounded by the two half-lines which are called the legs of the angle). Or, if we take an arbitrary circle and an angle whose vertex lies in the center of the circle, then the intersection of these two figures is a figure called a circular sector.

Let us add two more examples from the field of arithmetic: the sum of the set of all positive numbers and of the set of all negative numbers is the set of all numbers different from 0; the intersection of the set of all even numbers and of the set of all prime numbers is the set having as its sole element the number 2, this number being the only even prime number.

The addition and multiplication of classes are governed by various laws. Some of these are completely analogous to the corresponding theorems of arithmetic concerning the addition and multiplication of numbers—and it is for this very reason that the terms “addition” and “multiplication” have been chosen for the above operations; as an example we mention the COMMUTATIVE and ASSOCIATIVE LAWS of addition and multiplication of classes:

For any classes K and L, K ∪ L = L ∪ K and K ∩ L = L ∩ K.

For any classes K, L and M, K ∪ (L ∪ M) = (K ∪ L) ∪ M
and K ∩ (L ∩ M) = (K ∩ L) ∩ M.

The analogy with the corresponding arithmetical theorems becomes evident when we replace the symbols “∪” and “∩” by the usual signs of addition and multiplication, “+” and “·”.

Other laws, however, deviate considerably from those of arithmetic; the LAW OF TAUTOLOGY constitutes a characteristic example:

For any class K, K ∪ K = K and K ∩ K = K.

This law becomes obvious on reflecting upon the meaning of the symbols “K ∪ K” and “K ∩ K”; if, for instance, one adds to the elements of the class K the elements of the same class, one does not really add anything, and the resulting class is again the same class K.

We want to mention one other operation, which differs from those of addition and multiplication inasmuch as it can be performed, not on two classes, but only on one class. It is the operation which consists in forming, from a given class K, the so-called COMPLEMENT OF THE CLASS K, that is, the class of all things not belonging to the class K; the complement of the class K is denoted by:

K′.

If K, for instance, is the set of all integers, all fractions and irrational numbers belong to the set K′.

As examples of laws which concern the concept of complement and establish its connection with concepts considered earlier, we give the following two statements:

For every class K, K ∪ K′ = ∨.

For every class K, K ∩ K′ = ∧.

The first of these is called the class-theoretical LAW OF EXCLUDED MIDDLE, and the second the class-theoretical LAW OF CONTRADICTION.

The relations between classes and the operations on classes with which we have just become acquainted, and also the concepts of the universal class and the null class, are treated in a special part of the theory of classes; since the laws concerning those relations and operations tend to have the character of simple formulas reminiscent of those of arithmetic, this part of the theory is known as the CALCULUS OF CLASSES.

26. Equinumerous classes, cardinal number of a class, finite and infinite classes; arithmetic as a part of logic

*Among the remaining concepts which form the subject of investigation of the theory of classes there is one group which deserves particular attention and which comprises such concepts as equinumerous classes, cardinal number of a class, finite and infinite classes. They are, unfortunately, rather involved concepts which can only be superficially discussed here.

As an example of two EQUINUMEROUS or EQUIVALENT CLASSES, we may consider the sets of the fingers of the right and of the left hands; these sets are equinumerous, because it is possible to pair off the fingers of both hands in such a manner that (i) every finger occurs in just one pair, and (ii) every pair contains just one finger of the left hand and just one finger of the right hand. In a similar sense, the following three sets, for instance, are equinumerous: the set of all vertices, the set of all sides, and the set of all angles of a polygon. Later, in Section 33, we shall be able to give an exact and general definition of this concept of equinumerous classes.

Now let us consider an arbitrary class K; there exists, no doubt, a property belonging to all classes equinumerous to K and to no other classes (namely, the property of being equinumerous with K); this property is called the CARDINAL NUMBER, or the NUMBER OF ELEMENTS, or the POWER OF THE CLASS K. This can also be expressed more briefly and precisely, though perhaps in an even more abstract manner: The cardinal number of a class K is the class of all classes equinumerous with K. It follows from this that two classes K and L have the same cardinal number if, and only if, they are equinumerous.

With regard to the number of their elements, classes are classified into finite and infinite ones. Among the former, we distinguish between classes consisting of exactly one element, of two, of three elements, and so on. These terms are most easily definable on the basis of arithmetic. Indeed, let n be an arbitrary natural number (that is, a non-negative integer); then we shall say that THE CLASS K CONSISTS OF n ELEMENTS, if K is equinumerous with the class of all natural numbers less than n. In particular, a class consists of 2 elements, if it is equinumerous with the class of all natural numbers less than 2, i.e., to the class consisting of the numbers 0 and 1. Similarly, a class consists of 3 elements if it is equinumerous with the class containing the numbers 0, 1 and 2 as elements. In general, we shall call a class K FINITE if there exists a natural number n such that the class K consists of n elements, otherwise INFINITE.

It has, however, been recognized that there is still another possible procedure. All the terms which have just been considered can be defined in purely logical terms, without resorting at all to any expressions belonging to the field of arithmetic. We may, for instance, say that the class K consists of exactly one element, if this class satisfies the following two conditions: (i) there is an x such that ; (ii) for any y and z, if and , then y = z (these two conditions may also be replaced by a single one: “there is exactly one x such that ”; cf. Section 20). Analogously, we can define the phrases: “the class K consists of two elements”, “the class K consists of three elements”, and so on. The problem becomes much more difficult when we turn to the question of defining the terms “finite class” and “infinite class”; but also in these cases the efforts of solving the problem positively have been successful (cf. Section 33), and thereby all the concepts under consideration have been included within the range of logic.

This circumstance has a most interesting consequence of far-reaching importance; for it turns out that the notion of number itself and likewise all other arithmetical concepts are definable within the field of logic. It is, indeed, easy to establish the meaning of symbols designating individual natural numbers, such as “0”, “1”, “2”, and so on. The number 1, for instance, can be defined as the number of elements of a class which consist of exactly one element. (A definition of this kind seems to be incorrect and contains apparently a vicious circle, since the word “one”, which is about to be defined, occurs in the definiens; but actually no error is committed because the phrase “the class consists of exactly one element” is considered as a whole and its meaning has been defined previously.) Nor is it hard to define the general concept of a natural number: a natural number is the cardinal number of a finite class. We are, further, in a position to define all operations on natural numbers, and to extend the concept of number by the introduction of fractions, negative and irrational numbers, without, at any place, having to go beyond the limits of logic. Furthermore, it is possible to prove all the theorems of arithmetic on the basis of laws of logic alone (with the qualification that the system of logical laws must first be enriched by the inclusion of a statement which is intuitively less evident than the others, namely, the so-called AXIOM OF INFINITY, which states that there are infinitely many different things). This entire construction is very abstract, it cannot easily be popularized and does not fit into the framework of an elementary presentation of arithmetic; in this book we also do not attempt to adapt ourselves to this conception and treat numbers as individuals and not as properties or classes of classes. But the mere fact that it has been possible to develop the whole of arithmetic, including the disciplines erected upon it—algebra, analysis, and so on—, as a part of pure logic, constitutes one of the grandest achievements of recent logical investigations.⁴^*

Exercises

1. Let K be the set of all numbers less than ; which of the following formulas are true:

2. Consider the following four sets:

(a) the set of all positive numbers,

(b) the set of all numbers less than 3,

(d) the set of all numbers x satisfying the sentential function “x < 2x”.

Which of these sets are identical, and which are distinct?

3. What name is given in geometry to the set of all points in space whose distance from a given point (or from a given straight line) does not exceed the length of a given line segment?

4. Let K and L be two concentric circles, the radius of the first being smaller than that of the second. Which of the relations discussed in Section 20 holds between these circles? Does the same relation hold between the circumferences of the circles?

5. Draw two squares K and L so that they stand in one of the following relations:

(a) K = L,

(b) the square K is a part of the square L,

(d) the squares K and L overlap,

(e) the squares K and L are disjoint.

Which of these cases are eliminated, (i) if the squares are congruent, or (ii) if not the squares but only their perimeters are considered?

6. Let x and y be two arbitrary numbers, with x < y. It is well known that the set of numbers which are not smaller than x and not larger than y is called the interval with the endpoints x and y; it is denoted by the symbol “[x, y]”.

Which of the formulas below are correct:

(a) [3, 5] ⊂ [3, 6],

(b) [4, 7] ⊂ [5, 10],

(d) [−7, 1] ⊃ [−5, −2] ?

Which of the fundamental relations hold between the intervals:

(e) [2, 4] and [5, 8],

(f) [3, 6] and ,

(g) and ?

7. Is the following sentence (which has the same structure as the laws of syllogism given in Section 24) true:

if K is disjoint from L and L disjoint from M, then K is
disjoint from M ?

8. Translate the following formulas into terms of ordinary language:

(a) ,

(b) .

What laws mentioned in Section 22 and 24 find their expression in these formulas? What alterations on both sides of the equivalence (b) would be required in order to arrive at a definition of the symbol “⊂” or “⊃”?

9. Let ABC be an arbitrary triangle, with an arbitrary point D lying on the segment BC. What figures are formed by the sum of the two triangles ABD and ACD and by their product? Express the answer in formulas.

10. Represent an arbitrary square:

(a) as the sum of two trapezoids,

(b) as the intersection of two triangles.

11. Which of the formulas below are true (compare exercise 6):

(a) ,

(b) [−1, 2] ∪ [0, 3] = [0, 2],

(d) ?

In those formulas which are false correct the expression on the right of the symbol “=”.

12. Let K and L be two arbitrary classes. What classes are K ∪ L and K ∩ L in case K ⊂ L? In particular, what classes are K ∪ ∨, K ∩ ∨, ∧ ∪ L and ∧ ∩ L?

Hint: In answering the second question keep in mind a law of Section 24 concerning the classes ∨ and ∧.

13. Try to show that any classes K, L and M satisfy the following formulas:

(a) K ⊂ K ∪ L and K ⊃ K ∩ L,

(b) K ∩ (L ∪ M) = (K ∩ L) ∪ (K ∩ M) and K ∪ (L ∩ M) = (K ∪ L) ∩ (K ∪ M),

(d) (K ∪ L)′ = K′ ∩ L′ and (K ∩ L)′ = K′ ∪ L′.

The formulas (a) are called the LAWS OF SIMPLIFICATION (for addition and multiplication of classes); the formulas (b) are the DISTRIBUTIVE LAWS (for the multiplication of classes with respect to addition and for addition with respect to multiplication); the formula (c) is the LAW OF DOUBLE COMPLEMENT; and, finally, the formulas (d) are the class-theoretical LAWS OF DE MORGAN.⁵ Which of these laws correspond to theorems of arithmetic?

Hint: In order to prove the first of the formulas (d), for instance, it is sufficient to show that the classes (K ∪ L)′ and K′ ∩ L′ consist entirely of the same elements (cf. Section 24). For this purpose, we must, using the definitions of Section 25, make clear to ourselves when a thing x belongs to the class (K ∪ L)′ and when it belongs to the class K′ ∩ L′.

*14. Between the laws of sentential calculus given in Sections 12 and 13 and in Exercise 14 of Chapter II, on the one hand, and the laws of the calculus of classes given in Sections 24 and 25 and in the preceding exercise, on the other, there subsists a far-reaching similarity in structure (which is indicated in the analogy in their names). Describe in detail wherein this similarity lies, and try to find a general explanation of this phenomenon.

In Section 14, we became .acquainted with the law of contraposition of sentential calculus; formulate the analogous law of the calculus of classes.

15. With the help of the symbol:

introduced in Section 22 we can write the definition of the sum of two classes in the following way:

but it is also possible to restate this definition in the usual form of an equivalence (without the use of that symbol):

Formulate analogously in two ways the definitions of the universal class, of the null class, of the product of two classes, and of the complement of a class.

*16. Is there a polygon, in which the set of all sides is equinumerous with the set of all diagonals?

*17. Lay down definitions of the following expressions, using terms from the field of logic exclusively:

(a) the class K consists of two elements,

(b) the class K consists of three elements.

*18. Consider the following three sets:

(a) the set of all natural numbers greater than 0 and less than 4,

(b) the set of all rational numbers greater than 0 and less than 4,

Which of these sets are finite and which are infinite?

Give further examples of finite and infinite sets of numbers.

¹ The beginnings of the theory of classes—or, to be more exact, of that part of this theory which we shall denote as the calculus of classes below—are already found in G. BOOLE (cf. footnote 1 on p. 19). The actual creator of the general theory of sets as an independent mathematical discipline was the great German mathematician G. CANTOR (1845–1918); we are indebted to him, in particular, for the analysis of such concepts as equality in power, cardinal number, infinity and order, which will be discussed in the course of the present and the next chapters.—CANTOR’S set theory is one of those mathematical disciplines which are in a state of especially intensive development. Its ideas and lines of thought have penetrated into almost all branches of mathematics and have exerted everywhere a most stimulating and fertilizing influence.

² The concept of logical types introduced by RUSSELL is akin to that of the order of a class, and can even be conceived as a generalization of the latter,—a generalization which refers not only to classes but also to other things, for instance, to relations, which will be considered in the next chapter. The theory of logical types was systematically developed in Principia Mathematica (cf. footnote 1 on p. 19).

³ These relations were first investigated in an exhaustive manner by the French mathematician J. D. GERGONNE (1771–1859).

⁴ The fundamental ideas in this field are due to FREGE (cf. footnote 2 on p. 19); he developed them for the first time in his interesting book: Die Grundlagen der Arithmetik (Breslau 1884). FREGE’S ideas found their systematic and exhaustive realization in WHITEHEAD and RUSSELL’S Principia Mathematica (cf. footnote 1 on p. 19).

⁵ Cf. footnote 6 on p. 52.