ABSTRACT
The philosopher may he surprised, after all that has already been said concerning numbers, to find that he is only to learn about real numbers; and his surprise will be turned to horror when he learns that real is opposed to rational. 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. The rational numbers in order of magnitude form a series in which there is a term between any two. The rational numbers form a compact series. It is to be observed that, in a compact series, there are an infinite number of terms between any two, there are no consecutive terms, and the stretch between any two terms is again a compact series.
