The Berry Paradox comes in a few different forms. Here is one of the simplest examples of it for easy comprehension:
Name "the smallest possible integer NOT definable by fewer than twelve words".
It's easy to imagine examples that DON'T work:
the speed of light in meters per second [only 8 words]
the number of inches in a foot [7 words]
the number of stars in the universe [7 words]
one hundred thirty million, sixty seven thousand, three hundred and thirteen [11 words]
easy enough...
The problem arises however (if it isn't already obvious), that if you did concoct a sentence of 12 or more words to define some integer, and it IS the smallest such definable integer, IT can then be accurately designated (defined) by the original 11-word sentence above ("the smallest possible integer not definable by fewer than twelve words") -- thus a self-referential contradiction!
Another example of where mixing language/semantics with numbers/mathematics proves vexing, throwing light on illogical ambiguity within language.
More on the Berry paradox at Wikipedia:
http://en.wikipedia.org/wiki/Berry_paradox
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