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For subsets $A,B\subseteq \mathbb{N}$, a many-one reduction $A{\le }_{m}B$ is a total computable function $h$ with $n\in A⟺h(n)\in B$. A Turing reduction $A{\le }_{T}B$ is an oracle machine with oracle $B$ computing the total indicator function ${\chi }_A$. It may make several adaptive oracle queries. A many-one reduction gives a Turing reduction using one query; the definitions impose no enumerability assumption on $A,B$.

The Friedberg–Muchnik theorem asserts that there exist computably enumerable sets $A,B$ withWe give the finite-injury priority construction. Fix an effective list ${\Phi }_e$ of Turing functionals and imposein priority order $R_0,S_0,R_1,S_1,\dots$. Each strategy has an unassigned, waiting or protected state; a private witness $x$ when assigned; and a restraint on the oracle set when protected. All witnesses, even abandoned ones, are permanently recorded and never reused. A strategy for $R_e$ is the only strategy ever allowed to enumerate its witness into $A$; the analogous statement holds for $S_e$ and $B$.

Start with finite sets $A_0=B_0=\varnothing$. At stage $s$, inspect the first $s+1$ strategies. An unassigned strategy needs attention. A waiting $R_e$ strategy needs attention if the computation ${\Phi }_e^{B_s}(x)$ converges within $s$ steps; waiting $S_e$ is symmetric. Act on the highest-priority strategy needing attention, if any. On assignment choose a fresh witness greater than $s$, every earlier witness and every higher-priority restraint, and enter the waiting state. On a convergence with output $v$ and oracle use $u$, act as follows: if $v=0$, enumerate the witness into its own target set; if $v\ne 0$, leave the witness out. Enter the protected state and restrain the oracle below $u$. The current characteristic value at the witness is now different from $v$, including outputs other than zero or one. Whenever a strategy acts, initialize every lower-priority strategy, discarding its current witness and restraint but retaining the permanent record of used witnesses. If none needs attention, change nothing.

Every enumeration respects higher-priority restraints: its witness was chosen beyond them after its last initialization, and a later higher-priority action would have initialized it. Each stage is effective, finite and monotone in $A_s,B_s$, so $A={\bigcup }_s{A}_s$ and $B={\bigcup }_s{B}_s$ are computably enumerable. Lower-priority restraints may be violated by a higher-priority action, exactly the allowed injuries.

Induct on priority to verify finite injury and satisfaction. Once all higher-priority strategies have made their last actions, the current strategy receives its final assignment and can act at most once more, on a convergence. If convergence is observed it is then protected permanently; otherwise it waits without further action. In either case every strategy acts only finitely often. After the final initialization of $R_e$, if it acts on a convergence, no higher strategy subsequently changes its assumptions and every lower strategy respects its restraint. The observed oracle computation therefore remains valid for the final $B$, while the permanently private witness retains the opposite characteristic value in $A$.

If it waits forever, then ${\Phi }_e^B(x)$ cannot converge. A convergent final computation uses only finitely many oracle bits; those bits in the increasing enumeration $B_s$ eventually equal their final values. For a sufficiently large stage the same finite computation would have been observed, and after the higher strategies have stopped it would receive attention, contradicting perpetual waiting. Thus $R_e$ is satisfied either by disagreement or by divergence. The same proof satisfies every $S_e$. This is private-witness diagonalization for incomparable enumerable sets, with the finite-injury verification supplying the required permanent disagreements.

A computable set reduces to every oracle, so incomparability makes both sets noncomputable. Every computably enumerable set reduces to the halting problem; if either of these sets had the halting degree, the other would reduce to it. Thus their c.e. degrees are also incomplete, giving the intermediate degrees sought by the Post problem.