Hi,
In fact it is not quite hard to see that such a numbering exists : just take
any numbering of all Turing machines, say {\phi_i}, then remove all indices i
such that there is j<i such that \phi_j = \phi_i (equality is mathematically
correctly defined between functions from N to N).
Ah, and also, notice that what you call a "programming language" is nothing
else than an enumeration of all Turing machines.
However, if by "construction" you meant "recursive construction", i.e. for
another enumeration (\psi_i), find a Turing machine computing for any i, the
index j such that \phi_j=\psi_i, then it is clear that it is not possible : it
would allow to decide the equality problem for Turing machines.
Hope this helps...
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