Addendum

Among physicists, no one objected more violently to Cantorian set theory than Percy W. Bridgman. In Reflections of a Physicist (1955) he says he "cannot see an iota of appeal" in Cantor's proof that the real numbers form a set of higher infinity than the integers. Nor can he find paradox in any of Zeno's arguments because he is unable to think of a line as a set of points (see the Clerihew by Lindon that I used as an epigraph) or a time interval as a set of instants.

"A point is a curious thing," he wrote in The Way Things Are (1959), "and I do not believe that its nature is appreciated, even by many mathematicians. A line is not composed of points in any real sense. . . . We do not construct the line out of points, but, given the line, we may construct points on it. 'All the points on the line' has the same sort of meaning that the 'entire line' has. . . . We create the points on a line just as we create the numbers, and we identify the points by the numerical values of the coordinates."

Merwin J. Lyng, in The Mathematics Teacher (April 1968, page 393), gives an amusing variation of Black's moving-marble su-pertask. A box has a hole at each end: Inside the box a rabbit sticks his head out of hole A, then a minute later out of hole B, then a half-minute later out of hole A, and so on. His students concluded that after two minutes the head is sticking out of both holes, "but practically the problem is not possible unless we split hares."

For what it is worth, I agree with those who believe that paradoxes such as the staccato run can be stated without contradiction in the language of set theory, but as soon as any element is added to the task that involves a highest integer, you add something not permitted, therefore you add only nonsense. There is nothing wrong in the abstract about an ideal bouncing ball coming to rest, or a staccato moving point reaching a goal, but nothing meaningful is added if you assume that at each bounce the ball changes color, alternating red and blue; then ask what color it is when it stops bouncing, or if the staccato runner opens and shuts his mouth at each step and you ask if it is open or closed at the finish.

A number of readers called my attention to errors in this chapter, as I first wrote it as a column, but I wish particularly to thank Leonard Gillman, of the University of Texas at Austin for reviewing the column and suggesting numerous revisions that have greatly simplified and improved the text.

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