The string "seven" describes the number 7. So do the strings "four plus three", "7", and "the smallest prime number greater than 5".

Consider all English strings of 500 characters or less. Since there are a finite number of them, certainly they can only describe a finite number of numbers. Furthermore, they can only describe a finite number of positive integers. Since the number of positive integers is infinite, there are some that cannot be described in an English string of 500 characters or less.

Let us consider the set of all such integers. Since they are all positive, the set has a smallest member. Now consider the English string of 102 characters "the smallest positive integer that can't be described with an English string of 500 characters or less". So the smallest positive integer that can't be described with an English string of 500 characters or less can be described with an English string of 102 characters.

Peter Roosen-Runge has written a nice discussion / explanation of Berry's Paradox.