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Dull Numbers Become Cinderellas


In his article (below), Nick MacKinnon offers some humorous food for thought...that reveals some special aspects of mathematics.

A previous MathNEXUS entry offered a proof that all positive integers are interesting. MacKinnon suggests that this "classic poof" has some imprecision, and offers a correction.

First, he defines an "interesting number" to be any real number which is describable by a finite English phrase. And by black-and-white logic, a non-interesting number is considered "dull."

Then, he argues that every natural number is interesting, because of a similar proof by contradiction. That is, the smallest "dull" natural number would be describable by the finite English phrase "the smallest dull natural number."

MacKinnon then continues his argument by noting that the set of interesting real numbers are countable, which implies that the set of dull real numbers is uncountable (because the reals are an uncountable set).

Yet, as MacKinnon notes, no one will ever be able to state, list, or exhibit one of these dull real numbers...because then it would automatically become interesting. He concludes: "They sit in etyernal obscurity, Cinderellas who will never go to the ball."

Source: N. MacKinnon, "Interesting Real Numbers." Mathematical Spectrum, 1989/90, pp. 77-78.