and whole meet, and become in some wise precisely congruent, element for element.[1] We mention the other types of self-representation only to eliminate them from the present discourse.
In case of these self-representative systems, of the type especially interesting to us, we have already illustrated how their particular kind of self-representation developes infinite variety out of unity in a peculiarly impressive way. The general law of the process in question may now be stated, in a still more precise and technical form.
We may once more use the thoroughly typical case of the number-system. We have seen, in general, the positive nature of its endlessness. We want now to define, in decidedly general terms, the infinite process whereby the numbers can be self-represented, in infinitely numerous ways, by a part of themselves, and to state, abstractly, the implications of any such process. Let, then, f(n) represent any “function” of a whole number, such that n is to take, successively, the value of any whole number from 1 onwards; while f(n) itself is, in value, always a determinate whole number. The values of f(n) shall never be repeated. They shall follow in endless succession, and, as we shall also here suppose, in the order of their magnitude from less to more. Not all the numbers shall appear amongst the values of f(n). In consequence, f(n), by means of its first, second, third values, etc., shall represent precisely the whole of the number-series, while forming only a part thereof. Otherwise let f(n) be an arbitrary function. Then it will always be true that f(n) will contain, as a part of itself, a series f1(n), related to f(n) in precisely the same way in which f(n) is related to the original series of whole numbers. It
- ↑ Upon the various types of Ketten, finite and infinite, “cyclical” and “open,” see the very minute analysis given by Bettazzi, in his papers entitled Sulla Catena di un Ente in un gruppo, and Gruppi finiti ed infiniti di Enti, in the Atti of the Turin Academy of Sciences (for 1895-96), Vol. 31, pp. 447 and 506. Bettazzi, in the second of these papers, expresses some dissatisfaction with Dedekind’s definition of the Infinite, but withdraws his objections in a later paper, Atti, Vol. 32, p. 353.
