Cardinal Number. The cardinal number, then, of an assemblage a is something that is common to all the assemblages similar to a, or as we shall say, since classes are more amenable to mathematical treatment than properties, the class of all assemblages similar to a. We shall represent cardinal numbers by the letters ', P, V. ir, P, and the cardinal number of the as semblage a by the symbol Nc 'a. It may be shown that our ordinary 0 is en,and that l is Nc' ex, or the class of all the classes ex.
Finite and Infinite Cardinals. It is not universally true that the whole is greater than any of its parts. A picture one inch square may be enlarged to be two inches square. Each point of the two-inch picture, if the en largement be accurate, represents uniquely a point of the one-inch picture, and vice versa, yet the one-inch picture can be laid down on one of the corners of the two-inch picture. Obviously, therefore, there are as many points in one of the four quarters of a picture or a square plane area as in the entire picture or area Again, there is a first, a second., ...., an 'rib odd number, so that it is possible by con tinuing this process indefinitely to assign every odd number to an integer and every integer to an odd number, yet the odd numbers form only a part of the integers. Here are instances of assemblages no greater than their own parts. Inspection will show that no aggregate con taining a number of terms which can be ex pressed in the Arabic system of numeration can be of this type, for such aggregates are al ways reduced in number of terms when some are removed. The precise nature of these as semblages which can be enumerated by one of the ordinary numerals of the Arabic system will be discussed in a later paragraph. Here we call attention to the fact that it has not been demonstrated that all assemblages are either similar to one of their own parts or finite in the ordinary, every-day sense of having one of the natural numbers as their cardinal number. All apparent proofs of this depend on Zermelo's te axiom (See ASSEMBLAGES, GENERAL THEORY OF). However, no assemblages such as we should ordinarily call finite are similar to their own proper parts, so that assemblages of the latter sort and theii cardinal numbers are properly called infinite.
Greater and Less Among Cardinal Num bers. X is said to be greater than it if X is the cardinal number of some class a which contains a part fi of which is is the cardinal number, and if the converse relation does not hold. It is clearly impossible for X to be greater than and for is to be simultaneously p greater than X. That of any two cardinals one is theeater has not yet been demonstrated in though its proof for the natural num bers is easy. Thus while the natural numbers form a series, the infinites may constitute a tree or an anastomosing network. Here again
Zermelo's axiom underlies all the earlier work on the subject.
Addition. Consider the two numbers X and a. Let X be the cardinal number of the assemblage a and is that of R. Suppose that a and r: have no member in common: that is, that ant is A. It can then be shown that the cardinal number of a up is independent of the particular value of a and of chosen. Ex pressed in terms of X and a, Nc'
is written X + a, and is called the sum of X and P. As cabbages and kings are the same as kings and cabbages as a w /3 is identically R u a it follows that A + /A = + X, or in other words that the operation of addition, when per formed on any two cardinal numbers, finite or infinite, is commutative. In a similar way it may be shown that X + + v) =
The Natural Numbers. The *natural numbers* or *numbers that can be expressed in the Arabic notation' have already been men tioned.. The best definition of these has been given by A. N. Whitehead and B. Russell, to the effect that the natural numbers are those which are amenable to the method of mathe matical induction. That is, the natural num bers are those which possess every property which 1 possesses, which, when it is possessed by will also belong to A + 1, where + and I are used in the senses already defined. In other words, the validity of the method of mathematical induction as applied to the natural numbers rests neither on an axiom nor on a theorem, but on the definition of the natural numbers.
Given two assemblages a and R, we can derive from them the assem blage of all the ordered pairs of entities the first member of which is an a while the second is R. For example, given a class of Christian names and a class of surnames, we can con struct therefrom a class of all complete names of two parts, such that the Chris tian name is to be found among the given Christian names, while the surname is to be i found among the given surnames. It is clearly in harmony with our normal use of terms to say that if there are P Christian names and v surnames in the two sets respectively, there are i X v complete names in the derived set. We shall generalize this and make it our definition of multiplication: p X v, the product of P and v, is the number of couples formed by first selecting a term from a class of v members and then selecting a term taken from a class of v members. On the basis of the definition alone, the associative and commutative laws may be shown to hold of multiplication among any numbers, finite or infinite. Furthermore, the distributive law for multiplication with respect to addition the law that p X (i. + 1r) (P X v) X Tr) is easy of universal proof in the algebra we have established.