Examples for incomparable semi-decidable but undecidable languages
Some undecidable r.e. sets are isomorphic to the halting problem, for example the halting problem itself. Others are not, for example the one constructed by Friedberg and Muchnik using the priority method, a complicated kind of diagonalization. This set is not easy to describe, though one could argue that its description is "explicit". For some references, check out the Wikipedia article on simple sets, or any lecture notes on recursion theory.