For each admissible numbering, pick at least one pair of programs (but not necessarily all, which is impossible anyway) where the first translates from a given admissible numbering to that one, and the second work in the opposite way.
Now define a language consisting of all the pairs of programs chosen by one such way, for all admissible numberings. Is there such a language being recursively enumerable?
In other words, is there a way to enumerate all possible, sane programming languages (in the form of compilers), in a Turing-machine?
It is not even recognizable whether the two programs defines an admissible numbering. But it doesn't matter much since we only need to get one pair of translators for each numbering.