One may begin with the case as Aristotle first stated it, in the Third Book of the Physics, and elsewhere. There can, indeed, exist a Reality that permits us, if we choose to number its parts, to distinguish within it what we call elements, in such wise that we can never end the process of numbering them. So space is for us capable of infinite, that is, of indefinite division, if you choose to try to take it to pieces. But such divisibility is a mere possibility. Space, if real, is not endlessly divided. It is only in potentia divisible so far as you please to conceive its parts. The limitless exists, therefore, only in potentia; λείπεται οὖν δυνάμει εἶναι τὸ ἄπειρον. For were space actually either made up of endless parts, or in such wise real as to be infinitely great, there would result the contradiction of an actually infinite number as the number of the parts of a real collection. But a number actually infinite is contradictory; for it then could not be counted; it would have no determinate size; it would possess no totality; and it would so be formless and meaningless. Again, were any one portion of the world’s material substance infinite, how could room be left for the other portions? Were the whole infinite, how could it be a whole at all? For any whole of reality is limited by its own form, and by the fact that, as an actual whole, it is perfectly determinate. The difficulty as to the infinite must be solved, then, by saying that what is real forms a definite and, for that reason, a finite totality; while within this totality there may be aspects which our thought discovers to be, in this or that respect, inexhaustible through any process of counting that follows some abstractly possible line of our own subjective distinctions or syntheses. We can say, of such aspects of the world, that you may go on as long as you please, in counting their special type of conceived complexities, without ever reaching the end. But this endlessness is potential only, and never actual.
These well-known Aristotelian considerations have formed the basis of every argument against the actual infinite in later thought. The special point of attack has, however, often shifted. In general, as the later arguments have repeatedly
