Page:The World and the Individual, First Series (1899).djvu/544

Rh in the series of values of f(n) the rth place, while f(p), or r, occupies the rth place in the original number-series, one can say, in general, that the successive values of f1(n) are numbers which occupy in f(n) places precisely corresponding to the places which the successive values of f(n) themselves occupy in the original number-series. Thus the first member of f1(n) is that one amongst the members of the series of values of f(n) whose place in that series of values corresponds to the place in the original series of whole numbers which was occupied by f(1). The second member of f1(n) is, even so, that one amongst the series of values of f(n) which occupies the place in that series of values which f(2) occupies in the original number-series. And, in general, if, to the whole number p, in the original number-series, there corresponded the number r, as the image of that number in the series called f(n), then this pth member of the series called f(n) will have, as its image or representative in f1(n), the number f(r), i.e. the value of f(n) when n = r. This number f(r) will constitute, of course, the pth member of f1(n), and will occupy, in the series called f(n), the very same relative place which f(p) occupies in the original number-series.

Precisely so, f1(n) contains, as a part of itself, its own image as it is in f(n) and also as it is in the original series. And this new image may be called f2(n); and so on without end. Hence, one process of self-representation inevitably determines an endless Kette altogether parallel to our series of maps within maps of England. The general structure and development of any self-representative system of the present type have now been not only illustrated, but precisely defined and developed. Self-representation, of the type here in question, creates, at one stroke, an infinite chain of self-representations within self-representations.