urged (quite in Aristotle’s spirit), the infinitely complex, it real, must be knowable only through some finished synthesis of knowledge. But a finished synthesis is inconsistent (so one affirms) with the endlessness of the series of facts to be synthesized; and hence an infinite collection, if it existed, would be unknowable. On the other hand, an infinite collection, if real apart from knowledge, could be conceived to be altered by depriving it of some, or of a considerable fraction, of its constituent elements. The collection thus reduced (so one has often argued) would be at once finite (since it would have lost some of its members) and infinite, since no finite number would be equal to exhausting the remaining portion. Hence the reduced collection and, therefore, the original collection must be of a contradictory nature, and so impossible. In a variation of this argument often used, one employs, as an image, some such instance as an inextensible rod, one end of which shall be in my hands, while I shall be supposed to believe that the rod, which stretches out of my sight into the heavens, is infinitely long, as well as quite incapable of being anywhere stretched. Suppose the rod hereupon drawn, or, if you please, anyway mysteriously moved, a foot towards me at this end. If I am to believe in the infinity and inextensibility of the rod, I shall believe that the whole of the rod, and every part thereof, is now a foot nearer to me than before. But in that case the furthest portion of the rod must also be a foot nearer than before, or must have been “drawn in out of the infinite,” as one writer has stated the case.[1] It can therefore no longer be an infinite rod. Hence, it was not actually infinite before the drawing in of this end.
All such arguments insist, either upon the supposed fact
- ↑ Constantin Gutberlet, Zeitschrift für Philosophie (Ulrici-Falckenberg), Bd. 92, Hft. II, p. 199. The wording of the example is a little different in the text cited. The force of the argument no longer exists for one who approaches the concept of the Infinite through that of the Kette. Cantor observes as much in his answer to Gutberlet in the same journal. The puzzle turns upon falsely identifying the properties of finite and infinite quantities.
