Page:The World and the Individual, First Series (1899).djvu/535

516 whole number r, however large. Then, in the ideal class oF objects called whole numbers, there is a determinate even number which occupies the rth place in the series of even numbers, when the latter are arranged according to their sizes, beginning with 2. There is equally a prime number, occupying the rth place in a similarly ordered series of primes; and a square number occupying the rth place in a similarly ordered series of square numbers; and a cube occupying the rth place in a like arrangement of cubes; and an rth member in any particular series of numbers of the form ɑn, where n is any determinate whole number, and a is taken, in succession, as 1, 2, 3, etc. As all these things hold true for any r, however large, we can say, in general, that every whole number r has its correspondent rth member in any of the supposed series of systematically selected whole numbers, — even numbers, primes, square numbers, cubes, or what you will. But these various selected systems are such that each of them forms only a portion of the entire series of whole numbers. So that the whole series, taken as given, is in infinitely numerous ways capable of being put in a one-to-one relation to one of its own constituent parts.

I doubt not that this very fact might appear, at first blush, to bring out a manifest “contradiction” in the very conception of the “totality” of the whole numbers taken as “given.” But closer examination will show, as Couturat, Cantor, and the other authors here concerned (since Bolzano) have repeatedly pointed out, that the “contradiction” in question is really a contradiction only of the well-known nature of any finite collection. It was of such collections that the axiom, “The whole is greater than the part,” was first asserted. And of such collections alone is it with absolute generality true. Take any finite collection of whole numbers, however large; and then indeed the assertion of any of the foregoing one-to-one correspondences of the whole, with a mere part of itself, breaks down. But let us once see that taking any number r, however large, we can find the corresponding rth member in any of the ordered series of primes, squares, etc., and then we shall also