is always the same, whatever inductive number
may be. There is not any inductive cardinal to correspond to
. We may call it "the infinity of rationals." It is an instance of the sort of infinite that is traditional in mathematics, and that is represented by "
." This is a totally different sort from the true Cantorian infinite, which we shall consider in our next chapter. The infinity of rationals does not demand, for its definition or use, any infinite classes or infinite integers. It is not, in actual fact, a very important notion, and we could dispense with it altogether if there were any object in doing so. The Cantorian infinite, on the other hand, is of the greatest and most fundamental importance; the understanding of it opens the way to whole new realms of mathematics and philosophy.
[15]Of course in practice we shall continue to speak of a fraction as (say) greater or less than 1, meaning greater or less than the ratio
. So long as it is understood that the ratio