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 because it is the smallest uninteresting number, producing a contradiction. This contradiction continues as the next smallest number becomes the “smallest uninteresting number,” therefore becoming “interesting.”

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, the smallest integer that does not, as of November 2011, appear in an entry of the On-Line Encyclopedia of Integer Sequences was 12407.) 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.

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. 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.

via Interesting number paradox

Hi Jim, interesting article. Reminds me of Russell’s paradox which is also formed through self reference. Just to point out that the real numbers do form a well ordered set. One can validly say for instance “3.14 < pi 1) you cannot form an ordering though you can still form sets.

It appears to have chopped off half my comment… oh well. Should read “With complex numbers you cannot form an ordering etc etc”.

Hi Dan, thanks for that – very interesting. I did put a post on last year about Russell’s Paradox http://deskarati.com/2011/11/01/russells-paradox/ but I don’t think anybody got it. I also posted one about the great man himself http://deskarati.com/2011/08/04/bertrands-wise-words/ but don’t tell Alfy, or he will only knock down another one of my heroes!!