Aristotle's Wheel Paradox

A paradox mentioned in the Greek work Mechanica, dubiously attributed to Aristotle. Consider the above diagram depicting a wheel consisting of two concentric circles of different diameters (a wheel within a wheel). There is a 1:1 correspondence of points on the large circle with points on the small circle, so the wheel should travel the same distance regardless of whether it is rolled from left to right on the top straight line or on the bottom one. this seems to imply that the two circumferences of different sized circles are equal, which is impossible.

The fallacy lies in the assumption that a one-to-one correspondence of points means that two curves must have the same length. In fact, the cardinalities of points in a line segment of any length (or even an infinite line, a plane, a three-dimensional space, or an infinite dimensional Euclidean space) are all the same:  (the cardinality of the continuum), so the points of any of these can be put in a one-to-one correspondence with those of any other.

REFERENCES

Ballew, D. "The Wheel of Aristotle." Math. Teacher 65, 507-509, 1972.

Costabel, P. "The Wheel of Aristotle and French Consideration of Galileo's Arguments." Math. Teacher 61, 527-534, 1968.

Drabkin, I. "Aristotle's Wheel: Notes on the History of the Paradox." Osiris 9, 162-198, 1950.

Gardner, M. Wheels, Life, and other Mathematical Amusements. New York: W. H. Freeman, pp. 2-4, 1983.

Hutton, C. Mathematical and Philosophical Dictionary, Vol. 1. London: J. Davis, p. 398, 1795.

Pappas, T. "The Wheel of Paradox Aristotle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 202, 1989.

vos Savant, M. The World's Most Famous Math Problem. New York: St. Martin's Press, pp. 48-50, 1993.