Berry’s Paradox

After somewhat of a hiatus caused by continuously jumping from place to place over the summer, I wanted to come back to this blog. Today, instead of some physics I wanted to talk about a philosophical problem called Berry’s paradox. This is a paradox I came across some time ago in a philosophy lecture, though unfortunately I can’t remember by who. What surprised me is that it seemed to be being discussed as if it was a real problematic issue. A quick look on wikipedia will show you that the resolution is already known, but it will probably not enlighten you very much unless you already know quite a bit of the philosophical context. So today I wanted to share the way the resolution occurred to me when I heard of this paradox.

The following formulation of the Berry paradox can be found on Wikipedia:

Consider the expression:

“The smallest positive integer not definable in under sixty letters.”

Since there are only twenty-six letters in English alphabet, there are finitely many phrases of under sixty letters, and hence finitely many positive integers that are defined by phrases of under sixty letters. Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under sixty letters. If there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property; therefore, there is a smallest positive integer satisfying the property “not definable in under sixty letters”. This is the integer to which the above expression refers. The above expression is only fifty-seven letters long, therefore it is definable in under sixty letters, and is not the smallest positive integer not definable in under sixty letters, and is not defined by this expression. This is a paradox: there must be an integer defined by this expression, but since the expression is self-contradictory (any integer it defines is definable in under sixty letters), there cannot be any integer defined by it.

Like many paradoxes it does not so much uncover a deep philosophical problem, so much as a sloppy way with language (a prejudice of mine which would no doubt please Wittgenstein). The problem is that there is no consistent use of the word ‘definable’ in the paradox. If I am to take its formulation seriously then I have to assume that every single 59 letter string of the 26 latin letters of the English alphabet are used to refer to some number. In fact the above formulation does not stipulate the latin alphabet, but let’s assume that that is what is meant, it makes no difference to the argument. So perhaps we could have, with the letters on the left and the number represented on the right, (picking the integers represented at will):

a	1
aa	2
aaa	3
raspberry pie	31
strawberry pie	314

We would also need to assign a number to such strings as

fhdjkfhdsjflskjfdsl	1000
jhdfkjdslfjldsfmklds	1001

and inevitably to such sentences as

The smallest positive integer not definable in under sixty letters	30009
The smallest elephant under the bed	3008

Note this would include such phrases as “twenty one”, “einundzwanzing”, “vingt et un”, and so on. Though none of these need to refer to the number 21. In general it does not matter which integer refers to which string of letters, though we might try and construct a simple rule to do this job. The important point is that once one has finished, the phrase “The smallest positive integer not definable in under sixty letters” already refers to a number, and it certainly does not have to refer to the number described by that string in English. Just as the German sentence does not refer to that number, or indeed the sentence translated into any other language which can be transcribed into the latin alphabet. Paradox over. The mistake is to assume that the strings of letters must refer to the number that they describe in English.