Another part of the difficulty in computing this particular set comes from the indexing of theorems. No matter what indexing method our turing machine uses to index the theorems of peano arithmetic we could always construct a theorem that doesn't appear on the list our turing machine is to test. Just as cantor was able to show with the real #s.

I don't think that is correct. Cantor's ability to find real numbers that do not show up anywhere in any particular indexing scheme in which the integers are used to index the reals is dependent on the fact that the exact representation of many real numbers requires a string of digits of infinite length. This is not true of theorems in Peano arithmetic; although there are an infinite number of theorems in Peano arithmetic, each theorem can be represented with a string of symbols of finite length. So, there are countably many theorems and therefore we can index them using the integers.