less series of parts within parts. Each of these contained parts represents a preceding part precisely in the way in which the first representative system represents the original system. The law of the process always is that in a self-representative system of the type here in question, if any part A’ can stand in a one-to-one relation to the elements of the whole system, A, then ipso facto there exists A’’ (a part of this part), such that A’’ is the image or representative in A’, of A’ as it was in A. A’’ stands, then, in the same relation to A’, as that in which A’ stands to A; and A’’ is also a part of A. To derive A’ from A, by any such process as the one just exemplified, is therefore at once to define, by recurrence, the derivation of A’’ from A’, or, if you please, the internal and representative presence of A’’ within A’, of A’’’ within A’’, and so on without end. Nor can any A’ be derived from A, in such wise as exactly to represent, while a part of A, the whole of A, without the consequent implied definition of the whole series, also endless, A, A’, A’’, A’’’, wherein each term is a representative of the former term. So that not only is A self-representative and endless, but each of the derived series is self-representative and endless, while the whole ordered system of series that one can write in the orderly sequence A, A’, A’’, A’’’ is again a self-representative sequence, and so on endlessly, — all this complexity resulting self-evidently from the expressions of a single purpose.
One sees, — self-representation of the present type remains
persistently true to its tendency to develope types of variety
out of unity. Trivial these types may indeed seem; yet
the simplicity and the exactness of the derivation here in
question will soon prove of use to us in a wholly different field.
But it is now time to suggest, briefly, a still more general view
of these self-representative systems.
IV. Remarks upon the Various Types of Self-Representative Systems
We have so far spoken, repeatedly, of the “present type” of self-representative systems, meaning the type that, in this
