Page:The World and the Individual, First Series (1899).djvu/551

532 single element present in its predecessor. And the series of these successively omitted elements has an order absolutely predetermined by the one original plan. That order consists simply in the fact that each element omitted, when any of the new representations, N’’, N’’’, etc., is considered, is, upon each occasion, itself the Bild, the image or map or representative, of the very element that was previously omitted, when N’’, or N’’’, or other representation, was made. The endless series One, Two, Three, etc., is consequently the series of names of those objects whereof the first was omitted when the first representation or the mapping of N was made; while the second element represented, in the first map, N’, this first element of N. In the same way, in the second map, N’’, the element Three, the third element of the series, represented or pictured the second element, which latter, present in N’, had been omitted in N’’.

Thus the one plan of mapping or representing N by a part of itself, taken as a single act, accomplished at a stroke, logically involves what one can then express as an endless series of maps or images of the portion or element of N that is omitted from the first of the maps. And this endless ordered series of images of the omitted element of N, can be so carried out as to constitute a derived system that contains, in its turn, any member of N that you please, in a particular place, whose order in the series of successive images is absolutely predetermined by the one original plan. Hence, as Dedekind has it, “we say that the system N is, by this mode of representation, set in order (geordnet).” But, let us observe, this whole order, in all its infinite serial complexity, is logically accomplished by means of one act.

The series of images, or representations, of the element One, thus obtained, has of course, at first sight, a very artificial seeming. But a glance at the concrete case of the Gedankenwelt will show the sense of the process more directly. Let my Gedankenwelt be viewed in its totality, as a system self-represented in the way first defined. Then the one plan