The object of this paper is to explain a certain type of construction which occurs in priority proofs and illustrate it with two examples due to Lachlan and Harrington. The proofs in the examples are essentially the original proofs; our main contribution is to isolate the common part of these proofs. The key ideas in this common part are due to Lachlan; we include several improvements due to Harrington, Soare, Slaman, and the author.Our notation is fairly standard. If X is (...) an r.e. set, Xs is the finite set of elements enumerated in X before step s. if Ф is a recursive functional, Фs is the approximation to Ф at step s; it only queries the oracle about numbers xi and iless-than-or-equals, slantx; and right-pointing angle bracketx0,...,xk−1right-pointing angle bracket is an increasing function of each of its arguments. Sets are sometimes identified with their characteristic functions. Wj isthe jth r.e. set. Xc is the complement of X. (shrink)

We formulate an abstract version of the finite injury method in the form of the Baire category theorem. The theorem has the following corollaries: The Friedberg-Muchnik pair of recursively enumerable degrees, the Sacks splitting theorem, the existence of a minimal degree below 0′ and the Shoenfield jump theorem.