ABSTRACT
The series of real numbers, as ordinarily defined, consists of the whole assemblage of rational and irrational numbers, the irrationals being defined as the limits of such series of rationals as have neither a rational nor an infinite limit. This definition, however, introduces grave difficulties, which will be considered in the next chapter. For my part I see no reason whatever to suppose that there are any irrational numbers in the above sense; and if there are any, it seems certain that they cannot be greater or less than rational numbers. When mathematicians have effected a generalization of number they are apt to be unduly modest about it-they think that the difference between the generalized and the original notions is less than it really is. We have already seen that the finite cardinals are not to be identified with the positive integers, nor yet with the ratios of the natural numbers to 1, both of which express relations, which the natural numbers do not. In like manner there is a real number associated with every rational number, but distinct from it. A real number, so I shall contend, is nothing but a certain class of rational numbers. Thus the class of rationals less than ½ is a real number, associated with, but obviously not identical with, the rational number ½. This theory is not, so far as I know, explicitly advocated by any other author, though Peano suggests it, and Cantor comes very near to it.* My grounds in
favour of this opinion are, first, that such classes of rationals have all the mathematical properties commonly assigned to real numbers, secondly, that the opposite theory presents logical difficulties which appear to me insuperable. The second point will be discussed in the next chapter; for the present I shall merely expound my own view, and endeavour to show that real numbers, so understood, have all the requisite characteristics. It will be observed that the following theory is independent of the doctrine of limits, which will only be introduced in the next chapter.
