Interesting number paradox

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The interesting number paradox is a semi-humorous paradox that arises from attempting to classify numbers as "interesting" or "dull". 1729, for example, may be called 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). Any hypothetical partition of natural numbers into interesting and dull sets seems to fail. For instance, suppose all the numbers up to 37 are considered interesting, but 38 is not. That very property (38 being the first dull number) could well be considered an interesting fact, and hence 38 is not dull after all. In this way one may argue, using proof by contradiction that each natural number is unable to be the smallest dull number, and so one may declare that all numbers must be interesting. This conclusion contradicts the intuitive notion that the natural numbers are apparently too numerous for all of them to be subjectively interesting.

Since the above argument relies on a subjective, intuitive notion of "interesting", it should be understood as a half-humorous application of self-reference in order to obtain a paradox. However, as there are many significant results in mathematics that make use of self-reference (such as Godel's Incompleteness Theorem), the paradox illustrates some of the power of self-reference, and thus touches on serious issues in many fields of study.

This version of the paradox applies only to the natural numbers, as it depends on mathematical induction, which is only applicable to sets that are well-ordered; the argument does not apply to the real numbers. However, it would apply to the real numbers together with a fixed well-ordering.

The obvious weakness in the 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).