The set of theorems of PA is computably enumerable, but the set of theorems and anti-theorems (things provably false) is not computably separable, IE there is no computation which returns “true” for theorems, “false” for anti-theorems, and always returns some answer (we don’t care specifically what that answer is for anything that’s not a theorem or anti-theorem).
The set of theorems of PA is computably enumerable, but the set of theorems and anti-theorems (things provably false) is not computably separable, IE there is no computation which returns “true” for theorems, “false” for anti-theorems, and always returns some answer (we don’t care specifically what that answer is for anything that’s not a theorem or anti-theorem).