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The false assumption underlying berry's paradox1

Published online by Cambridge University Press:  12 March 2014

James D. French
James D. French*
Affiliation:
Institute of General Semantics, Englewood, New Jersey 07631
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 4 , December 1988 , pp. 1220 - 1223
Copyright
Copyright © Association for Symbolic Logic 1988

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Footnotes

1

The task of writing this paper was greatly facilitated by methods of evaluation acquired from the study of Alfred Korzybski's educational discipline. I am also grateful to Jon Barwise, who made several important suggestions, and to the referee, who suggested the phrase “non-context-dependent,” among other changes.

References

REFERENCE

[ 1 ] Barwise, Jon and Etchemendy, John, The liar: an essay on truth and circularity, Oxford University Press, Oxford, 1987.Google Scholar
[2] Burge, Tyler, Semantical paradox, Recent essays on truth and the liar paradox (Martin, R. L., editor), Oxford University Press, Oxford, 1984, pp. 83–117.Google Scholar
[3] Parsons, Charles, The liar paradox, Recent essays on truth and the liar paradox (Martin, R. L., editor), Oxford University Press, Oxford, 1984, pp. 9–45.Google Scholar
[4] Pinter, Charles C., Set theory, Addison-Wesley, Reading, Massachusetts, 1971.Google Scholar