On Classes in General

1. A class (Menge, ensemble) is said to be determined by any test or condition which every entity (in the universe considered) must either satisfy or not satisfy; any entity which satisfies the condition is said to belong to the class, and is called an element of the class. A null or empty class corresponds to a condition which is satisfied by no entity in the universe considered.

For example, the class of prime numbers is a class of numbers determined by the condition that every number which belongs to it must have no factors other than itself and 1. Again, the class of men is a class of living beings determined by certain conditions set forth in works on biology. Finally, the class of perfect square numbers which end in 7 is an empty class, since every perfect square number must end in 0, 1, 4, 5, 6, or 9.

2. If two elements a and b of a given class are regarded as interchangeable throughout a given discussion, they are said to be equal; otherwise they are said to be distinct. The notations commonly used are a = b and a ? b, respectively.

3. A one-to-one correspondence between two classes is said to be established when some rule is given whereby each element of one class is paired with one and only one element of the other class, and reciprocally each element of the second class is paired with one and only one element of the first class.

For example, the class of soldiers in an army can be put into one-to-one correspondence with the class of rifles which they carry, since (as we suppose) each soldier is the owner of one and only one rifle, and each rifle is the property of one and only one soldier.

Again, the class of natural numbers can be put into one-to-one correspondence with the class of even numbers, since each natural number is half of some particular even number and each even number is double some particular natural number; thus:

1,2,3, ...,

2,4,6, ...

Again, the class of points on a line AB three inches long can be put into one-to-one correspondence with the class of points on a line CD one inch long; for example by means of projecting rays drawn from a point O as in the figure.

4. An example of a relation between two classes which is not a one-to-one correspondence, is furnished by the relation of ownership between the class of soldiers and the class of shoes which they wear; we have here what may be called a two-to-one correspondence between these classes, since each shoe is worn by one and only one soldier, while each soldier wears two and only two shoes. The consideration of this and similar examples shows that all the conditions mentioned in the definition of one-to-one correspondence are essential.

5. Obviously if two classes can be put into one-toone correspondence with any third class, they can be put into one-to-one correspondence with each other.

6. A part ("proper part," echter Teil), of a class A is any class which contains some but not all of the elements of A, and no other element.

A subclass (Teil) of A is any class every element of which belongs to A; that is, a subclass is either a part or the whole.

7. We now come to the definition of finite and infinite classes.

An infinite class is a class which can be put into oneto-one correspondence with a part of itself. A finite class is then defined as any class which is not infinite.

This fundamental property of infinite classes was clearly stated in B. Bolzano's Paradoxien des Unendlichen (published posthumously in 1850), and has since been adopted as the definition of infinity in the modern theory of classes.

8. An example of an infinite class is the class of the natural numbers, since it can be put into one-to-one correspondence with the class of the even numbers, which is only a part of itself (§ 3).

Again, the class of points on a line AB is infinite, since it can be put into one-to-one correspondence with the class of points on a segment CD of AB (by first putting both these classes into one-to-one correspondence with the class of points on an auxiliary line HK, as in the figure).

The class of the first n natural numbers, on the other hand, is finite, since if we attempt to set up a correspondence between the whole class and any one of its parts, we shall always find that one or more elements of the whole class will be left over after all the elements of the partial class have been assigned (see § 27).

9. The most important elementary theorems in regard to infinite classes are the following:

(1) If any subclass of a given class is infinite then the class itself is infinite.

For, let A be the given class, A' the infinite subclass, and A? the subclass of all the elements of A which do not belong to A' (noting that A? may be a null class).

By hypothesis, there is a part, A', of A' which can be put into one-to-one correspondence with the whole of A'; therefore the class composed of A'and A? will be a part of A which can be put into one-to-one correspondence with the whole of A.

(2) If any one element is excluded from an infinite class, the remaining class is also infinite.

For, let A be the given class, x the element to be excluded, and B the class remaining. By hypothesis, there is a part, A1, of A, which can be put into one-to-one correspondence with the whole of A, and is therefore itself infinite. If this part A1 does not contain the element x, it will be a subclass in B, and our theorem is proved. If it does contain x, there will be at least one element y which belongs to B and not to A1, and by replacing x by y in A1 we shall have another part of A, say A2, which will be an infinite part of A and at the same time a subclass in B.

10. As a corollary of this last theorem we see that no infinite class can ever be exhausted by taking away its elements one by one.

For, the class which remains after each subtraction is always an infinite class, by § 9, 2, and therefore can never be an empty class, or a class containing merely a single element (these classes being obviously finite according to the definition of § 7).

This result will be used in § 27, below, where another, more familiar, definition of finite and infinite classes will be given.

The further study of the theory of classes as such, leading to the...