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.
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[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 not interesting numbers. Being the smallest number of a set one might consider not interesting makes that number interesting. Since the numbers in this set were defined as not interesting, we have reached a contradiction because this smallest number cannot be both interesting and uninteresting. Therefore the set of uninteresting numbers must be empty, proving 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. (The paradox is alleviated if "interesting" is instead defined objectively: for example, as of November 2008, the smallest natural number that does not have its own Wikipedia entry is 215.) 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 irony 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).
[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)
- Mike Keith: All Numbers Are Interesting: A Constructive Approach
