ABSTRACT
One of the mistakes that have delayed the discovery of correct definitions in this region is the common idea that each extension of number included the previous sorts as special cases. The infinity of rationals does not demand, for its definition or use, any infinite classes or infinite integers. It is clear that fractions can be found which approach nearer and nearer to having their square equal to 2. It is clear that an irrational Dedekind cut in some way “represents” an irrational. Just as ratios whose denominator is 1 are not identical with integers, so those rational numbers which can be greater or less than irrationals, or can have irrationals as their limits, must not be identified with ratios.
