## Berry’s paradox

### August 29, 2012

At the moment I’m reading ‘The Information‘ by James Gleick. I’m only halfway through and it’s a pretty good read so far, though a lot more verbose than Chaos (his first book). It’s sort of ironic that a book about information would have a lower information density than a book about chaos.

When Gleick talks about Russell and famous Set Theory paradoxes, he briefly touches upon the Berry paradox [page 179-180].

It has to do with counting the syllables needed to specify each integer. Generally, of course, the larger the number the more syllables are required. In English, the smallest integer requiring two syllables is se·ven. The smallest requiring three syllables is e·le·ven. The number 121 seems to require six syllables (“one·hun·dred·twen·ty·one”), but actually four will do the job, with some cleverness: “e·le·ven·squared”. Still, even with cleverness, there are only a finite number of possible syllables and therefore a finite number of names, and, as Russell put it, “Hence the names of some integers must consist of at least nineteen syllables, and among these there must be a least. Hence

the least integer not nameable in fewer than nineteen syllablesmust denote a definite integer.”[…] Now comes the paradox. This phrase,the least integer not nameable in fewer than nineteen syllables, contains only eighteen syllables. So the least integer not nameable in fewer than nineteen syllables has just been named in fewer than nineteen syllables.^{[syllable notation mine]}

It took me a while to figure out what was meant by this being a paradox rather than just a linguistic trick. In fact my girlfriend —who is a linguist— still thinks it *is* a trick, so maybe I’m confused about this still. My understanding is that it is a paradox because of the word “not”. If there would be an integer that can*not* be described in less than 19 syllables, then seemingly it *can* be described in merely 18 after all. Therefore there cannot be a smallest integer which cannot be described in fewer than 19 syllables, which in turn means that all the integers between zero and infinity can be described using the permutations of a finite number of syllables in a limited length sequence, which is clearly bunk. Thus; paradox.

The Wikipedia page claims the resolution is due to ambiguous language, which is pretty much what girlfriend has been trying to tell me as well. I’m clearly missing something yet as I don’t understand how that solves anything.

September 2, 2012 at 11:37

Either one of the editors for Wikipedia have written that paradox wrong, or you (and it would seem you’re not alone in that respect) have misinterpreted the Berry Paradox (you should thank the confusion caused by the misinterpretation from the wiki editor again for that; see below for an explanation why). To me it isn’t a paradox at all, rather a very subtle play with words that serves only to stupify rather then challenge the intellect. Here’s my take, and please forgive me if i happen to be misguided:

The statement of “The smallest positive integer not definable in under 11 words” according to wikipedia means: it is definable in under eleven words / and is not the smallest positive integer not definable in under eleven words / and is not defined by this expression.

The first statement here is wrong which invalidates the other two. My reasoning? Read the wording of the paradox very carefully. The “smallest” positive integer “not” definable in “under” 11 words, actually contains a hidden double-negative. Instead it is simply asking the following: “what is the smallest positive integer definable in 11 words”. The original statement is not -as wiki and most other people interpret- declaring that the number ‘is’ under 11 words. It can’t be; by the inclusion of “‘not’ definable in under 11 words” would declare that any integer below eleven words is invalid, so it has to be 11 words minimum. Hence the solution is as I’ve pointed out and isn’t a paradox at all:

“what is the smallest positive integer definable in 11 words”

September 26, 2012 at 01:04

How about that : http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems ?

November 18, 2013 at 22:55

The paradox of “The smallest positive integer not definable in under 11 words” is that this statement has 10 syllables.

July 18, 2014 at 18:34

I don’t see a vicious circularity? If it just contradicts itself then how is it a paradox?