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  "body": "\n## Definition\n\nA *finite-injury priority construction* builds sets $A \\subseteq \\N$ by a computable process in stages $s = 0, 1, 2, \\ldots$, as follows.\n\n1. The goal is a list of *requirements* $R_0, R_1, R_2, \\ldots$, and $R_i$ has *higher priority* than $R_j$ when $i \u003c j$.\n2. 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|TN9G24VABN]].\n3. At a stage, an action taken for $R_i$ may undo the work done for some $R_j$ with $j \u003e i$; then $R_j$ is *injured* and starts again.\n4. An action for $R_j$ never injures any $R_i$ with $i \u003c j$.\n\nThe construction is *finite-injury* if each requirement is injured only finitely often and acts only finitely often.\nThe *priority method* is the use of such constructions to prove the existence of computably enumerable sets with given properties.\nIt also covers extensions in which requirements may be injured infinitely often.\n\n## Remarks\n\nExample: the proof of the [[Friedberg–Muchnik theorem|RA358M9QCN]] is a finite-injury construction with requirements $A \\ne \\Phi_e^B$ and $B \\ne \\Phi_e^A$.\nBy induction on $i$, after some stage no requirement before $R_i$ acts, so $R_i$ is injured only finitely often and is met.\n\nNon-example: a construction of $A$ that at each stage asks whether a given program halts is not a priority construction in this sense.\nIt is not a computable process, so it does not make $A$ computably enumerable.\n\nThe proof that a finite-injury construction succeeds is usually an induction on the priority order.\nInfinite-injury methods, such as those used for the Sacks density theorem, allow some requirements to be injured infinitely often.\n",
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