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". The paradox states that all numbers are interesting; the "proof" is by contradiction: if there were uninteresting numbers, there would be a smallest uninteresting number - but the smallest uninteresting number is itself interesting, producing a contradiction.

1 Proof
2 Paradoxical Nature
3 See also
4 Further reading

[edit] Proof

Claim: There is no such thing as an uninteresting number.

Proof by Contradiction

Assume that you have a set of numbers that are not interesting. Order the numbers from smallest to largest, then the smallest number in that set of uninteresting set of numbers is interesting since it is the smallest uninteresting number. Repeat this process until you are left with only one number in this uninteresting set and then that number is interesting since it is the only uninteresting number. Hence there is no such thing as an uninteresting number.

[edit] Paradoxical Nature

Attempting to classify all numbers this way leads to a paradox or an antinomy of definition. Any hypothetical partition of natural numbers into interesting and dull sets seems to fail. Since the definition of interesting is usually 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 Gödel'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 well-ordered sets with a natural order, such as 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 natural and fixed well-ordering.

One obvious weakness in the proof is that we have not properly defined the predicate of "interesting". But assuming this predicate 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 with particular opinions on numbers: if one's view is that all numbers are boring, and one finds uninteresting the observation that 0 is the smallest boring number, there's no paradox.

Another weakness in the proof is the assumption that what is interesting now was indeed interesting previously or will be interesting in the future (that "intrinsic interestingness" does not change). Yet another weakness is the dynamic nature of the solution but the static nature of the presentation of the problem (wherein lies the humor of the paradox). The proof essentially has the same weaknesses as most paradoxes/proofs of changing referents (e.g., the ontological proof of the existence of God).