Home > Interesting number paradox

At first sight, some numbers seem to be more "interesting" than others: 1729, for example, has the interesting property that it is the smallest number expressible as the sum of two positive perfect cubes in two different ways.

However, attempting to classify numbers as either "dull" and "interesting" leads to a paradox (strictly speaking, an antinomy). In a classification of numbers as to whether they had interesting properties or not, there would be a smallest number with no interesting properties (for instance, 38 could be a candidate). This in itself would be an interesting property of the number, so it would no longer be dull. This is a classic paradox of self-reference or proof by contradiction.

Consideration shows that the smallest or largest dull number is required to be a surreal number which cannot be mathematically defined. These dull numbers are, in essence, so dull that nobody knows what they are, nor cares what they might be.

Other interesting paradoxes involving dull numbers are:

• each number in the set of dull numbers is interesting for not being in the set of interesting numbers
• each number in the set of interesting numbers is dull, because there are many other interesting numbers

One of the problems with this proof is that we have not properly defined the predicate of "interesting". But assuming this predicate is defined, is defined with a finite, definite list of "interesting properties of positive integers", and is defined self-referentially to include the smallest number in such a list, a paradox (strictly speaking, an antinomy of definition) arises. The Berry paradox is closely related, arising from a similar self-referential definition.

It may be noted that some dull numbers necessarily have fewer uninteresting properties than others, as human knowledge is countable and finite. While there may be infinitely many unknown properties, these properties are interesting due to being unknown. Consequently, there exists a set of the most dull numbers (having only one uninteresting property) as well as a set of the least dull numbers (having the largest subset of the uninteresting properties). These sets are therefore interesting. Note that no dull number may have zero properties, as having zero properties is itself a property (though if such a number was found, it would undoubtably be extremely interesting).

1 See also

• Church-Turing Thesis
• Gödel's incompleteness theorem
• Hardy-Ramanujan number

2 Further reading

• Martin GardnerMartin Gardner (born October 21, 1914) is an American recreational mathematician and author of the long-running but now discontinued Mathematical Games' column in Scientific American. Interests and writings Born in Tulsa, Oklahoma, Martin Gardner more or, Mathematical Puzzles and Diversions, 1959Events January-February January 1 Cultivars of plants named after this date must be named in a modern language, not in Latin. January 1 Cuba: Fulgencio Batista flees Havana when forces of Fidel Castro advance January 2 CBS Radio cuts four soap operas: Bac