Beyond Infinity

To show that the real numbers cannot be counted, Cantor first established a fact which, if anything, seems to be almost beyond belief: There are as many points along an infinite straight line as there are on a finite segment of it. The proof, shown in Fig. 10.1, is so simple that one wonders why no one before him had made the discovery. It shows that our conception of a line as being made up of many dots of ink is fundamentally wrong: the physical dot and the mathematical point have absolutely nothing in common!