Here “separation” would mean that there is an algorithm which inputs any statement and outputs either “yes” or “no”, such that the algorithm returns “yes” on all inputs that are provable statements and “no” on all inputs that are disprovable statements. But the algorithm also has to halt on all possible inputs, not just the ones that are provable or disprovable. Such a separation algorithm cannot exist (I am not sure if this follows from Gödel’s theorem or requires a separate diagonalization argument). This is the result needed in that step of the argument.
Here “separation” would mean that there is an algorithm which inputs any statement and outputs either “yes” or “no”, such that the algorithm returns “yes” on all inputs that are provable statements and “no” on all inputs that are disprovable statements. But the algorithm also has to halt on all possible inputs, not just the ones that are provable or disprovable. Such a separation algorithm cannot exist (I am not sure if this follows from Gödel’s theorem or requires a separate diagonalization argument). This is the result needed in that step of the argument.
Ah, so I was quite wrong when I said “trivially they can be separated”. Cos we only have semi-decision procedures for provability and disprovability!
Thanks for helping me with this