Priority method

Definition

A finite-injury priority construction builds sets $A \subseteq \N$ by a computable process in stages $s = 0, 1, 2, \ldots$, as follows.

1. The goal is a list of requirements $R_0, R_1, R_2, \ldots$, and $R_i$ has higher priority than $R_j$ when $i < j$.
2. At each stage $s$, finitely many numbers are added to the finite approximation $A_s$, so $A_0 \subseteq A_1 \subseteq \cdots$ and $A = \bigcup_s A_s$ is computably enumerable.
3. At a stage, an action taken for $R_i$ may undo the work done for some $R_j$ with $j > i$; then $R_j$ is injured and starts again.
4. An action for $R_j$ never injures any $R_i$ with $i < j$.

The construction is finite-injury if each requirement is injured only finitely often and acts only finitely often. The priority method is the use of such constructions to prove the existence of computably enumerable sets with given properties. It also covers extensions in which requirements may be injured infinitely often.