Interesting number paradox
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The interesting number paradox arises from attempts to classify numbers as "interesting" or "dull". 1729, for example, is often considered an interesting number because it is the smallest number expressible as the sum of two positive perfect cubes in two different ways.
However, attempting to classify all numbers this way leads to a paradox (strictly speaking, an antinomy of definition). In a classification of numbers as to whether they had interesting properties or not, there would be a smallest number with no interesting properties. For instance, all the numbers up to 37 may have interesting properties, but 38 might not. This in itself would be an interesting property of the number 38, so it would no longer be dull. This is a classic paradox of self-reference or proof by contradiction.
This version of the paradox applies only to the natural numbers, as it depends on the set of numbers under consideration being well-ordered. (The real numbers, for example, are not well-ordered using the standard ordering ≤, and indeed there are a great many uninteresting real numbers, such as 197.3341.)
One of the problems with this proof is that we have not properly defined the predicate of "interesting". But assuming this predicate is defined, is defined with a finite, definite list of "interesting properties of positive integers", and is defined self-referentially to include the smallest number not in such a list, a paradox arises. The Berry paradox is closely related, arising from a similar self-referential definition. As the paradox lies in the definition of "interesting", it applies only to persons of sufficiently sophisticated taste in numbers. If one's view is that all numbers are boring, and one is unmoved by the observation that 0 is the smallest boring number, there's no paradox.
It may be noted that some dull numbers necessarily have fewer uninteresting properties than others. While there may be infinitely many unknown properties, these properties are interesting due to being unknown. So there are finitely many dull properties. Consequently, there exists a set of the most dull numbers (having the most uninteresting properties) as well as a set of the least dull numbers (having the fewest uninteresting properties). These sets are therefore interesting. Note that no dull number may have zero properties, as having zero properties is itself a property (if such a number were found, it would undoubtedly be extremely interesting).
See also
Further reading
- Martin Gardner, Mathematical Puzzles and Diversions, 1959 (ISBN 0226282538)fr:Paradoxe des nombres intéressants