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**