We should define this more formally. Twenty-six letters case-insensitive plus space, commas, and periods. Then we should start from 0 and find a many example as we can.
"Smallest natural number that cannot be defined by n characters [a-z ,.]"
0:1
1:1
2:1
3:3
4:3
Not following the bottom line, but sure. I think getting worried over the details misses the point. I think you'd like this version better:
Let A be the set of all positive integers that can be defined in under 100 words. Since there are only finitely many of these, there must be a smallest positive integer n that does not belong to A. But haven't I just defined n in under 100 words?
From here, where you can read a bit more
No, I understand the paradox. I just want to know the answer once you remove the paradox. If I change 100 to 10 then the paradox goes away and there is some number that meets the criteria. I want to know what it is!
isn't it just the finite field represented by the set of symbols and places?
No. You can easily define numbers larger than the set. Which means the largest number you can describe with words is somewhere within the set.
you can only define numbers larger than the set if you allow the set to be expanded in some way. more symbols, more places.
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?
