Regular readers of TSZ will remember the hilarity that ensued when former commenter JoeG grappled unsuccessfully with the cardinality (loosely, the size) of various infinite sets. In honor of that amusing episode, I’m posing a new problem involving an infinite set.

Here’s the problem:

Consider the set containing every real number that can be described using a finite number of English words. For example, “thirty-three” and “two point eight” obviously qualify as members of the set, but also “pi minus six”, “the cube root of e”, and “Zero Mostel’s age in years on July seventh, nineteen sixty-three”, all of which designate specific real numbers. The set is infinite, of course.

Prove that the set of all such numbers takes up exactly zero percent of the real number line.