Quantitative relations between infinite sets

Summary

Given the old conception of the relation greater than, the proposition that the whole is greater than the part is an immediate consequence. But being greater in this sense is not incompatible with being equal in the sense of one-one correspondence. Some who failed to recognize this formulated invalid arguments against the possibility of infinite quantities. Others who did realize that the relations of equal and greater when so defined are compatible, concluded that such relations are not appropriately taken as quantitative relations, at least, not in general. If suitable quantitative relations were to be defined, there was a possibility of retaining either the traditional definition of greater and finding another concept of equality, or of retaining the concept of equality as one-one correspondence and defining greater than so that it is incompatible with equality in this sense. The former alternative was implicitly taken by Bolzano, who never succeeded in defining suitable quantitative relations. The latter alternative was taken by Cantor, and is the basis of his great success in constructing a mathematical theory of the transfinite.