Berry’s paradox

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 syllables must 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 cannot 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.