Berry's paradox involves describing something using a description whose form is in apparent contradiction with the meaning of the description. As Bertrand Russell put it::
"'The least integer not nameable in fewer than nineteen syllables' is itself a name consisting of eighteen syllables; hence the least integer not nameable in fewer than nineteen syllables can be named in eighteen syllables, which is a contradiction." [Russell, B., "Mathematical Logic is Based on the Theory of Types", Am. J. of Mathematics 30, 1908, p. 223.]
The paradoxical force:
"Paradoxes are talismans. . . A good paradox can never be finally disposed of. . . The very existence of a paradox such as this can be used to derive facts about the relationship between the mind and the universe." [Rucker, R., Infinity and the Mind, Bantam, 1983, pp. 101.]
A proposed therapy :
"Deny that sufficiently rich languages can formalize the concept of denotation for their terms." (Thomas Tymoczko, New Directions in the Philosophy of Mathematics, Birkhauser: 1985, p. 288.)
Berry's paradox is an example of a family of supposed paradoxes which have in fact no paradoxical force and require no therapy. They typically involve defining a function which one might naively think is a total function on a certain domain but which turns out to be onlypartial. In the case of Berry's paradox, the partial function is the function which is supposed to assign (positive) integers to some domain of objects we call "phrases". Since the intended interpretation is that the phrases in question refer to specific integers, we will call the function the meaning function: meaning(x).
The purpose of the following discussion is to explain why the meaning function must be partial.
For this, we will need to define a couple of additional functions. One is thelength of the phrase. This function should have the property that only a finite number of phrases have a given length; otherwise, it doesn't matter how the length is computed. The function is also required to be defined for every phrase; it is a total function.
The second function we will call the Berry function:
Berry(n, meaning , length) = smallest integer m such that m >= meaning(x) for all x with length(x) < n.
In other words, the Berry function of n gives us the smallest integer whose description (if there is one) requires a phrase of length greater than or equal to n. This seems well-defined since there are only a finite number of phrases with length < n, and m is just
1 + max (meaning(x) : length(x) < n).
(You may wish to check at this point that you have been following the argument by answering the question: Why does m require a description of length >= n, and why is it the smallest which does?)
But giving a definition of a function does not mean a function is actually defined. In this case, the Berry function is only defined for n if meaning(x) is defined for all phrases x of length less than n. Conversely, if the Berry function is partial, undefined for some n, then the meaning function must be as well, for all the other operations and functions in the definition of Berry are total.
Now from the definitions, we have immediately: for no meaning and length functions is it the case that for some x shorter than n
Berry(n, meaning , length) = meaning(x).
Putting it another way, for any given x and n,
either length(x) >= n or
meaning(x) <> Berry(n, meaning, length) or
Berry(n, meaning, length) is undefined or
meaning(x) is undefined.
Thus, if we can construct a situation in which meaning(x) is defined for all phrases shorter than n (so that Berry(n, meaning, length) is defined) then it will follow that if meaning(x) is defined at all,
meaning(x) <> Berry(n, meaning, length).
We are led into paradox by believing that the meaning and Berry functions are somehow independent, that somehow since the meaning function was 'given beforehand', it couldn't be affected by the 'subsequent' definition of the Berry function. Yet now it seems that the Berry function has punched a "hole" in the meaning function. The description of the value Berry(n, meaning, length) seems perfectly unambiguous, even if it is a bit complicated, and now we are told that no matter what we take to be the domain of phrases, no phrase can mean that value? Surely the phrase "Berry(n, meaning, length)" is just such a phrase.
A crucial example: suppose
Don't we have a contradiction here? For doesn't
meaning("Berry( 25, meaning, length)" ) = Berry( 25, meaning, length)
contradict the result above? No, the equality is simply false. We cannot fix the meaning of the phrase "Berry( 25, meaning, length)" without at the same time determing the values of the Berry function which, in turn, depends on the meaning function. In this case, if the meaning function is defined for all phrases shorter than 25, then Berry( 25, meaning, length) will be 1 more than the maximum of the meanings of those phrases, so that the maximum can't be
Berry( 25, meaning, length)
itself, since that phrase has a length less than 25, But what prevents it? Simply that the meaning function fails to be defined for some phrase shorter than 25, in particular, the phrase
"Berry( 25, meaning, length)".
But suppose we think of the meaning function as a computational procedure which evaluates phrases. Then, surely, all the meaning function has to do is to strip off the quotation marks and evaluate the expression Berry( 25, meaning, length) ? Doesn't this define its value for this phrase? No, since the evaluation of the expression must be non-terminating! It cannot terminate in a value, since the supposition that it does yields a contradiction.
Again, the paradoxical force stems from the apparent oddity that by choosing meaning to be total, we have forced the Berry function, which is composed from meaning and other total functions such as + and max, to be partial. But all this means is that the choice was an illusion -- the meaning function is not total. That that strikes us as paradoxical is just due to our unfamiliarity with such situations; a very similar device is used in the standard proof that the halting problem for Turing machines is undecidable, and that result is no more paradoxical than this one is.
Tymoczko's linguistic therapy is not required. Languages can be quite rich without rendering them unable to formalize concepts of denotation (meaning) for their terms. The language we used to express the relation between a number and its description is an example. All that needs to be avoided in such languages, as in any other, is the claim that meaningful concepts can have contradictory properties. But this is a basic feature of the way logicians and mathematicians work; there is no paradox in that.