User blog:Alemagno12/Solution to Berry's Paradox

Let's say you made a theory T, and you made a function F such that F(n) is the largest number definable in T using at most n symbols. To calculate the values of F(n), we need to check each of the possible numbers definable in T using at most n symbols, then compare them. Let's say these are the results: We don't count gazillion, since it's ill-defined. So F(1) = 1, F(2) = 2, F(3) = 10, and F(4) = googol. But, we run into a problem when calculating F(5): We need to calculate F(5) in order to calculate F(5). So our theory T doesn't work...right?
 * 1 symbol: {1}
 * 2 symbols: {1,2}
 * 3 symbols: {1,2,10,4,3}
 * 4 symbols: {1,2,10,4,3,9,googol,-1,gazillion}
 * 5 symbols: {1,2,10,4,3,9,googol,-1,gazillion,9894832,decker,pi,e,F(5)}

Wrong. See, F(5) is ill-defined, since to calculate F(5), we have to calculate F(5), which causes an infinite loop. So we don't count F(5) when calculating F(5). So we get F(5) = decker. However, F(5) is not ill-defined now, because F(5) = decker. But this just repeats decker in the set of numbers definable in T using at most 5 symbols, so nothing happens.

Now, we run into another problem when calculating F(6): Now we have to calculate F(100) to calculate F(6). We can get two results: Option 1 is impossible, since every ill-defined value of F(n) will become well-defined, as shown in the calculation of F(5). So F(100) has to be well-defined. Then F(6) ≥ F(100). However, this means that F(n) doesn't have to be greater than F(m) if n is greater than m. This is a problem when defining a large number using F. However, to fix this problem, we can create a new function G, which is defined like this: So who's going to define the largest number definable in English using at most a googol words?
 * 6 symbols: {1,2,10,4,3,9,googol,-1,gazillion,9894832,decker,pi,e,F(5),10^4444444,boogol,tritri,googolminex,F(100)}
 * Option 1: F(100) is ill-defined, so we don't count F(100).
 * Option 2: F(100) is well-defined, so we count it.
 * G(1) = F(1)
 * G(n) (n > 1) = F(m) for the smallest possible value of m such that F(m) > G(n-1)