The false assumption underlying berry's paradox

Extract

This paper is divided into three sections. §1 consists of an argument against the validity of Berry's paradox; §2 consists of supporting arguments for the thesis presented in §1; and §3 examines the possibility of re-establishing the paradox.

Berry's paradox, a semantic antinomy, is described on p. 4 of the textbook [4] as follows:

For the sake of argument, let us admit that all the words of the English language are listed in some standard dictionary. Let T be the set of all the natural numbers that can be described in fewer than twenty words of the English language. Since there are only a finite number of English words, there are only finitely many combinations of fewer than twenty such words—that is, T is a finite set. Quite obviously, then, there are natural numbers which are greater than all the elements of T; hence there is a least natural number which cannot be described in fewer than twenty words of the English language. By definition, this number is not in T; yet we have described it in sixteen words, hence it is in T.

We are faced with a glaring contradiction; since the above argument would be unimpeachable if we admitted the existence of the set T, we are irrevocably led to the conclusion that a set such as T simply cannot exist.