Enumerate Turing recognizable language in lexicographic order

Question

So a Turing recognizable language is also enumerable.

But is it possible for an enumerator to "print" a Turing recognizable language in lexicographic order?

Answer 1

No. Suppose it were possible. Then the language could be decided as follows:

1. Begin enumerating all accepted strings in order.
2. If you list a string whose lexicographic order is greater than the string you're looking for, halt reject.
3. If you list the string you're looking for, halt accept.

This will eventually terminate for any input string since for any given input string only finitely many enumerated strings exist with smaller lexicographic order. It will always halt reject if it doesn't find your target string and it will always halt accept if it does. It doesn't have to worry about missing your string since it knows where your string should be, thanks to having listed strings in order.

Since we know there are enumerable languages that aren't decidable, it must be that we can't enumerate enumerable languages in order.