Simple sets were devised by Emil Leon Post in the search for a non-Turing-complete c.e.
Whether such sets exist is known as Post's problem.
Post had to prove two things in order to obtain his result: that the simple set A is not computable, and that the K, the halting problem, does not Turing-reduce to A.
Post's idea was validated by Friedberg and Muchnik in the 1950s using a novel technique called the priority method.
They give a construction for a set that is simple (and thus non-computable), but fails to compute the halting problem.