you can only define numbers larger than the set if you allow the set to be expanded in some way. more symbols, more places.

Sorry I got the backwards. "Smallest number you can't describe with words"

The symbols only have meaning if you give them meaning. Let's say I give you five places 0-9 and the lower case alphabet. So you write 'zzzzz' as the largest integer therefore '100000' must be the smallest integer that can't be defined right? Well I'd argue that 'zzzzz' doesn't define anything since it doesn't define its own base. What if you had written 'tree3'? Is that a base 36 or the usual huge number we normally define it as?