Chapter 3.4, Problem 20MS

The number name paradox. Let S be the set of all natural numbers that are describable in English words using no more than 50 characters (so, 240 is in S since we can describe it as “two hundred forty,” which requires fewer than 50 characters). Assuming that we are allowed to use only the 27 standard characters (the 26 letters of the alphabet and the space character), show that there are only finitely many numbers contained in S. (In fact, perhaps you can show that there can be no more than 27⁵⁰ elements in S.) Now, let the set T be all those natural numbers not in S. Show that there are infinitely many elements in T. Next, since T is a collection of natural numbers, show that it must contain a smallest number. Finally, consider the smallest number cont ained in T. Prove that this number must simultaneously be an element of S and not an element of S—a paradox!