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.

Paradoxical nature[edit]

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 appear in an entry of the On-Line Encyclopedia of Integer Sequences and was originally found to be 11630 on 12 June 2009.[1] The number fitting this definition later became 12407 from November 2009 until at least November 2011, then 13794 as of April 2012, until it appeared in sequence OEIS:A218631 as of 3 November 2012. Since November 2013, that number was 14228, at least until 14 April 2014.[1] (Note that this is only possible because the OEIS only lists a finite number of terms for each entry. For instance, OEIS:A000027 is the sequence of all natural numbers, and if continued indefinitely would contain all positive integers. As it is, the sequence is only recorded in its entry as far as 77.) 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.[2] 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.

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.[3] 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.

See also[edit]


  1. ^ a b Johnston, N. (June 12, 2009). "11630 is the First Uninteresting Number". Retrieved November 12, 2011. 
  2. ^ Charles R Greathouse IV. "Uninteresting Numbers". Retrieved 2011-08-28. 
  3. ^ Clark, M., 2007, Paradoxes from A to Z, Routledge, ISBN 0-521-46168-5.

Further reading[edit]

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