Editing Computability theory (section)

===The priority method===
{{further|Turing degree#Post's problem and the priority method}}
Post's problem was solved with a method called the ''priority method''; a proof using this method is called a ''priority argument''. This method is primarily used to construct computably enumerable sets with particular properties. To use this method, the desired properties of the set to be constructed are broken up into an infinite list of goals, known as ''requirements'', so that satisfying all the requirements will cause the set constructed to have the desired properties. Each requirement is assigned to a natural number representing the priority of the requirement; so 0 is assigned to the most important priority, 1 to the second most important, and so on. The set is then constructed in stages, each stage attempting to satisfy one of more of the requirements by either adding numbers to the set or banning numbers from the set so that the final set will satisfy the requirement. It may happen that satisfying one requirement will cause another to become unsatisfied; the priority order is used to decide what to do in such an event.

Priority arguments have been employed to solve many problems in computability theory, and have been classified into a hierarchy based on their complexity.<ref name="Soare_1987"/> Because complex priority arguments can be technical and difficult to follow, it has traditionally been considered desirable to prove results without priority arguments, or to see if results proved with priority arguments can also be proved without them. 
For example, Kummer published a paper on a proof for the existence of Friedberg numberings without using the priority method.