Turing degree
When Post defined the notion of a simple set as an c.e. set with an infinite complement not containing any infinite c.e. set, he started to study the structure of the computably enumerable sets under inclusion. This lattice became a well-studied structure. computable sets can be defined in this structure by the basic result that a set is computable if and only if the set and its complement are both computably enumerable. Infinite c.e. sets have always infinite computable subsets; but on the other hand, simple sets exist but do not always have a coinfinite computable superset. Post (1944) introduced already hypersimple and hyperhypersimple sets; later maximal sets were constructed which are c.e. sets such that every c.e. superset is either a finite variant of the given maximal set or is co-finite. Post's original motivation in the study of this lattice was to find a structural notion such that every set which satisfies this property is neither in the Turing degree of the computable sets nor in the Turing degree of the halting problem. Post did not find such a property and the solution to his problem applied priority methods instead; Harrington and Soare (1991) found eventually such a property.