Berry Paradox: a contradiction that arises in the description of very large numbers and presents a fundamental problem for the interpretation of language.
The Berry Paradox is based on an approach similar to that of ►Richard's Paradox: It seems obvious that all finite ►numbers should be uniquely describable (determinable) in language by means of a given finite number of letters. Thus, the number 13 is uniquely determined as "thirteen", "twelve plus one" and "the next ►prime after eleven". Even the quite impressive number "ten to the power of one hundred" requires no more than 26 letters for its linguistic identification. Yet, how many letters do we need to uniquely describe the following number?
The smallest number that can not be uniquely described in fewer than one hundred letters.
If you think that the answer is a hundred letters, then why don't you count the letters in the italicized phrase above? Obviously, it took less than hundred letters to uniquely describe the smallest number that can not be uniquely described in fewer than one hundred letters!
A librarian by the name of Berry first formulated this contradiction and passed it on to the philosopher/logician Bertrand Russell, who in turn went ahead and published it. You may think that this so-called paradox is only an apparent paradox that merely arises from the sloppiness of everyday language. Indeed, the above number definition (in italics) somehow looks fishy. Does it even suffice to uniquely describe a particular number? Well, the almost identically-worded definite description the smallest number that can not be uniquely described in fewer than six letters does, for each language, uniquely identify a specific number; thus, in English it would be the number 11.
Does the paradox get dissolved by translating it into the formal language of mathematics? To attempt this, let us define three mathematical ►functions:
The function Denotation determines, for any English numerical description, the described number. Thus, Denotation("seven") yields the number 7, Denotation("largest prime under twenty") yields 19. Denotation can yield the same number for different descriptions, but not different numbers for the same description.
Length yields the length of a denoting expression in terms of its number of letters, e.g., Length("eleven") = 6. The third function Berry is applied to a given number n to yield the smallest number whose shortest denoting expression has at least n letters -- in other words, the smallest number that is larger than all numbers with descriptions shorter than n, or in mathematical notation
Berry(n) = 1 + max(Denotation(s): Length(s) < n)*
As we saw, in English we have Berry(6) = 11, because the smallest number not uniquely describable in English using fewer than 6 letters is the number 11. Since there are many ways to uniquely describe one and the same number, Berry(n) is less easily determined for a very large n. Indeed, you can imagine that as the value for n increases, the values of our Berry function will soon defy all limits and, not unlike the ►Ackermann function, rapidly approach infinity.
Yet the function does seem to be clearly defined, for there is indeed only a finite number of descriptions with lengths less than n for any given n. However, the Berry definition is valid only if there are also clear definitions of Denotation(s) for all descriptions s with lengths less than n, and vice versa. This raises the question, Could there possibly be a description s for which Denotation(s) is undefined?
As we see from the above Berry definition, there could not be a description s shorter than n such that
Berry(n) = Denotation(s).
That is to say, for any given s and n, either
Length(s) ≥ n, or
Denotation(s) ≠ Berry(n), or
Berry(n) or Denotation(s) is undefined.
However, let us take a look at this:
Length("Berry(one hundred)") = 15
Denotation("Berry(one hundred)") = Berry(100)
Taken together, these two equations violate the restriction above; since the length of the description "Berry(one hundred)" is obviously less than 100, either Denotation("Berry(one hundred)") must yield a value different from the number Berry(100) or one of the functions Denotation and Berry is undefined for some descriptions with less than 100 letters. Even if we make the assumption that
Denotation("Berry(one hundred)") ≠ Berry(100)
we face serious difficulties trying to concretely determine the value of Berry(100). For this would require us to know the value of Denotation(s) for all s with less than 100 letters including that of Denotation("Berry(one hundred)"). But to determine the value of Denotation("Berry(one hundred)") we would first need to know the value of Berry(100). Thus the Berry function's definition is incomplete. For to determine its value we already need to know its value. And with discovering the incompleteness of Berry's number definition we have dissolved the paradox.
Hence, the description the smallest number that can not be uniquely described in fewer than one hundred letters does not suffice to uniquely identify one particular number. This is so even though there are only finitely many numbers that can be uniquely described using fewer than one hundred letters. It follows from Berry's paradox that there is no formal algorithm -- such as our above function Denotation -- to fully interpret any given language. This result has substantial consequences in various fields of research, including artificial intelligence (see ►soul).
A seemingly harmless linguistic curiosity at first, our paradox has turned out to be quite a troublemaker. Indeed, the Berry paradox has parallels to Turing's halting problem (see ►infinite loop) and Gödel's incompleteness theorem (see ►continuum hypothesis). In fact, Gödel's incompleteness theorem can be fully proven in terms of the Berry paradox.**
* max(a: b) = Largest of all a that are b.
** See for this G. J. Chaitin, Lecture given on October 27, 1993, at the University of New Mexico, http://www.cs.auckland.ac.nz/CDMTCS/chaitin/unm2.html