Georg Cantor's diagonal method is said to do an amazing thing. It is said to prove that, while the rational numbers are denumerable, the real numbers are not. Transfinite theorist consider this dichotomy between the rational and reals to be the basis of arithmetic, logic, calculus and all higher mathematics.
Unfortunately, the diagonal method falls short of its lofty goal, and often does little more than confuse students, and add to the epidemic of math anxiety plaguing our nation schools.
The standard presentation of the diagonal method is a deceptively simple three step lecture. In the first two steps are supposed to prove that the rational numbers are denumerable. The final step is supposed to show that the real numbers are not denumerable. The proof looks clean, but on closer examination, we find that the definition of "denumerable" is not as clear as we might hope.
The standard presentation of the diagonal method is deceptively simple. It focuses the students attention on the symbolic representation of the rational numbers, while diverting attention from the much more important issue of the definition of the central terms of the theory.
For this reason, I begin my essays with a parody of the diagonal method, and not with a discussion of the central terms of the theory. The parody contains no useful information, and you can skip right to the review of the diagonal method, if you desire.
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