User:Charles Matthews/interesting and uninteresting numbers

The issue of classifying numbers as "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, would there be a smallest number with no interesting properties (for instance, 38 could be a candidate)? If there was a finite, definite list of 'interesting properties of positive integers', which didn't exhaust all integers, there would be such a number.

This in itself would be an 'interesting' property of the number, making it interesting, thus excluding it from the list. The Berry paradox is closely related.

Of course the argument fails if there could be infinitely many such properties; or if interesting is not a very well-defined predicate.

The anecdote about 1729 has been cited in this connection. Here 1729 is the third Carmichael number. Would Hardy have really thought it "dull"? The 1729th decimal place is the beginning of the first occurrence of all ten digits consecutively in the decimal representation of e. This fact would not (presumably) have been known. It seems that over time what is 'interesting' may change, too.