User blog comment:LittlePeng9/First order oodle theory/@comment-30167082-20161222182647/@comment-24920136-20161223012129

the foot function is worded like so:

the largest natural number which can be defined in the language of FOOT in at most n symbols.

So you see  you are actually looking for the right combination and positioning of characters that could output the largest number. This works similarly to how  BB(n) does  for turing machines (although there are a few differences).

Ill give you a toy example: a language that only has natural numbers and the ^*+ symbols (exponentiation, addition, multiplication) lets call it "toy". Its trivial to show that toy can express any number. ie: 10 can be expressed as 1+1+1+1+1+1+1+1+1+1. or 2*5.

Its also obvious that the expressions represent a number uniquely, for example, 2*5 only expresses 10 and not any other number.

Toy(n) is then the largest number expressible in Toy in 5 characters

So for example toy(5) = 9^9^9

The wording in rayos function is a bit different, it says something close to " the smallest integer that is bigger than any number expressible in N characters by a fost expression "

so if i used that wording then it would make

toy(5) = (9^9^9)+1