Interesting number paradox

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The interesting number paradox is a semi-humorous paradox that arises from attempting to classify natural numbers as "interesting" or "dull". The paradox states that all natural 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.

Contents 1 Proof 2 Paradoxical nature 3 Notes 4 See also 5 Further reading

[edit] Proof

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

Proof by Contradiction: Assume that you have a non-empty set of natural numbers that are not interesting. Due to the well-ordered property of the natural numbers, there must be some smallest number in the set of uninteresting numbers. Being the smallest number of a set one might consider not interesting makes that number interesting after all: a contradiction.

[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. (The paradox is alleviated if "interesting" is instead defined objectively: for example, as of June 2009, the smallest natural number that does not have its own Wikipedia entry is 215, and the smallest number that does not appear in an entry of the On-Line Encyclopedia of Integer Sequences is 12407^[1].) 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; the argument would not apply to the real numbers.

One proposed resolution of the paradox asserts that only the first uninteresting number is made interesting by that fact. For example, if 39 and 41 were the first two uninteresting numbers, then 39 would become interesting as a result, but 41 would not since it is not the first uninteresting number.^[2]

An obvious weakness in the proof is that what qualifies as "interesting" is not defined. However, 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, since it arises 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 is no paradox.

[edit] Notes

1. ^ Johnston, N. (June 12, 2009). "11630 is the First Uninteresting Number". http://www.nathanieljohnston.com/index.php/2009/06/11630-is-the-first-uninteresting-number/. Retrieved June 16, 2009. 
2. ^ Clark, M., 2007, Paradoxes from A to Z, Routledge, ISBN 0-521-46168-5.

[edit] See also

• Grelling–Nelson paradox
• Richard's paradox
• Church-Turing thesis
• Gödel's incompleteness theorem
• Hardy–Ramanujan number
• List of paradoxes

[edit] Further reading

• Martin Gardner, Mathematical Puzzles and Diversions, 1959 (ISBN 0-226-28253-8)