Interesting number paradox
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The interesting number paradox is a semi-humorous paradox which arises from the attempt to classify natural numbers as "interesting" or "dull". The paradox states that all natural numbers are interesting. The "proof" is by contradiction: if there exists a non-empty set of uninteresting numbers, there would be a smallest uninteresting number – but the smallest uninteresting number is itself interesting because it is the smallest uninteresting number, producing a contradiction.
- Claim: There is no such thing as an uninteresting natural number.
- Proof by Contradiction: Assume that there is a non-empty set of natural numbers that are not interesting. There must be some smallest number in the set of uninteresting numbers. Being the smallest number of a set one may consider uninteresting makes that number interesting after all: a contradiction.
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, the smallest integer that does not, as of November 2011, appear in an entry of the On-Line Encyclopedia of Integer Sequences was 12407, but as of April 2012, was 13794 (as of 3 November 2012, appears in sequence OEIS:A218631.) Depending on the sources used for the list of interesting numbers, a variety of other numbers can be characterized as uninteresting in the same way. This might be better described as "not known to be interesting".
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.
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. However, this resolution is invalid, since the paradox is proved by contradiction: assuming that there is any uninteresting number, we arrive to the fact that that same number is interesting, hence no number can be uninteresting; its aim is not in particular to identify the interesting or uninteresting numbers, but to speculate whether any number can in fact exhibit such properties.
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.
- Church–Turing thesis
- Gödel's incompleteness theorems
- Grelling–Nelson paradox
- Hardy–Ramanujan number (An example of an interesting number)
- List of paradoxes
- Richard's paradox
- Unexpected hanging paradox
- Kleene–Rosser paradox
- Johnston, N. (June 12, 2009). "11630 is the First Uninteresting Number". Retrieved November 12, 2011.
- Charles R Greathouse IV. "Uninteresting Numbers". Retrieved 2011-08-28.
- Clark, M., 2007, Paradoxes from A to Z, Routledge, ISBN 0-521-46168-5.
- Gardner, Martin (1959). Mathematical Puzzles and Diversions. ISBN 0-226-28253-8.
- Gleick, James (2010). The Information (chapter 12). New York: Pantheon Books. ISBN 978-0-307-37957-3.