which, taken in its order, is the next element of the system. This next element may be viewed, if we choose, as derived from its predecessor by means of the recurrent process. But it may also be viewed as in a relation to its predecessor, which is the same as the relation of a map to an object mapped. We shall accordingly call it, henceforth, the Image or Representation of this former element. (2) These images are all distinct, so that various elements always have various representatives. For the recurrent process is such that, in the system which should finally express it, one and only one element would be derived from any given element, or would be the next element in order after that given element. (3) At least one element, M, of the system, although imaged by another, is itself the image or representative of no other element, so that only a portion of the system is representative. A system thus defined we may call, for our present purposes, an instance of an internally Self-Representative System, or, more exactly, of a system precisely represented by a proper fraction or portion of itself. Of the whole system thus defined we can at once assert that if we take its elements in the order M, M’, M’’, etc., there is indeed no last member in the resulting series. The system is, therefore, defined as endless merely by being defined as thus self-representative. But since the self-representation of any system of facts is capable of definition, as a single internal purpose, in advance of the discovery that such purpose involves an endless series of constituents, we may, with Dedekind, use the generalized conception of a self-representation of the type here in question as a means of positively defining what we mean by an infinite system or multitude of elements. In thus proceeding, we further generalize the idea which the perfect map of England has already illustrated.
The positive definition of the concept of the Infinite thus resulting has no small speculative interest. Ordinarily one defines infinity merely by considering some indefinitely prolonged series of successive facts, by observing that the series in question does not, or at least, so far as one sees, need not,
