This paradox, proposed by the French
logician Jules Antoine Richard, is summed up (hrm) with the phrase,
"The smallest number that cannot be defined by a phrase in the English language containing fewer than twenty words."
The
brute-force approach to defining such a number lies in assembling a
set of all possible phrases, each 19 or less words in length, that define a number, and finding the smallest number not defined by a phrase in the set. The paradox is that the statement itself is 19 words long, and thus is a member of the set it implies. Like
Jabberwocks and the square root of -1, Richard's Number must remain
imaginary.
Ian Stewart, in the January 2001
Scientific American ("Mathematical Recreations" sidebar, pg 103), proposes further problems with list items that refer to other list items, since their inclusion in the set may be situational. Consider the phrases:
"The number named in the next expression, if a number is named there, and zero if not." (17 words)
"One plus the number named in the preceding expression." (9 words)
Taken together in the order shown, neither phrase defines a number, and so they don't seem to be valid set members. With the right "neighbors", however, they would be valid expressions.
A Blather of Paradoxes