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For sets $A,B\subseteq \mathbb{N}$, many-one reducibility means there is a total computable function $h$ such thatTuring reducibility $A{\le }_{T}B$ means that a Turing machine with oracle $B$ computes the indicator function ${\chi }_A$, halting on every input. A many-one reduction supplies a Turing reduction by computing $h(x)$ and making one membership query, so $A{\le }_{m}B$ implies $A{\le }_{T}B$.

The Friedberg–Muchnik theorem states that there are computably enumerable sets $A,B$ with $A/{\le }_{T}B$ and $B/{\le }_{T}A$. We prove this by a finite-injury priority construction. Fix an effective enumeration $({\Phi }_e)$ of the oracle Turing functionals and arrange the requirements in the orderA requirement owns a private witness, a success flag and a priority restraint on its oracle set. All witnesses ever chosen are distinct. An $R_e$ witness $x$ is reserved outside $A$, and only $R_e$ may enumerate that number into $A$; an $S_e$ witness is treated symmetrically. A priority restraint $r$ forbids lower-priority requirements from adding numbers less than $r$ to the specified oracle set. Initializing a requirement abandons its witness, clears its success flag and sets its priority restraint to zero; enumeration into $A$ or $B$ is never undone.

At stage $s$ consider the first $s+1$ requirements and serve the highest-priority one requiring attention. An unsuccessful requirement requires attention if it has no witness, or its current witness has a stage-$s$ oracle computation with output zero, found within $s$ steps. If there is no such requirement, do nothing. When a requirement with no witness is served, choose a fresh number larger than $s$, all previously chosen witnesses and all current higher-priority priority restraints. When an $R_e$ with a witness $x$ and observed computation ${\Phi }_e^{B_s}(x)=0$ is served, enumerate $x$ into $A$, mark it successful, and restrain $B$ below the use of that computation. The use is one more than the largest queried oracle position, or zero if there were no queries. An $S_e$ acts in exactly the same way with $A,B$ interchanged. After either kind of service, initialize every currently assigned lower-priority requirement; the remaining lower priorities are already in their initial state.

This is an effective stage construction with finite sets $A_s,B_s$, and $A={\bigcup }_s{A}_s$, $B={\bigcup }_s{B}_s$ are computably enumerable. Every enumeration respects all higher-priority priority restraints: its private witness was selected above them, and any later service to a higher-priority requirement would have initialized it before it could act. We do not need to restrain a computation whose output is nonzero; a persistent nonzero output already disagrees with an unenumerated witness.

We verify the finite injury assertion by induction along the priority order. After the last service to requirements above a fixed requirement $Q$, the requirement $Q$ is never initialized again. It can be served at most twice thereafter, once to choose its final witness and once to diagonalize. Whenever it needs attention it is eventually served, since the higher-priority requirements have ceased acting and each later stage considers a longer initial segment. Thus $Q$ has a final witness and makes only finitely many actions; this proves the induction and justifies the assumed last service above it.

For a final $R_e$ witness $x$, if the diagonalization occurs then ${\chi }_A(x)=1$, while preservation of a restrained oracle computation applies: its zero computation is preserved permanently because higher priorities no longer act and lower priorities respect the priority restraint. Thus ${\Phi }_e^B(x)=0\ne {\chi }_A(x)$. If it never occurs, then ${\chi }_A(x)=0$. Were ${\Phi }_e^B(x)=0$, that convergent computation would use finitely many oracle answers. Because $B_s$ increases to $B$, those finitely many answers eventually stabilize, and a sufficiently late stage would discover the zero computation and cause diagonalization. This contradicts its never occurring. So in this case ${\Phi }_e^B(x)$ is undefined or nonzero and again disagrees with ${\chi }_A$. The identical argument verifies every $S_e$.

All the requirements hold, proving the theorem. Both sets are noncomputable: a computable set is Turing reducible to every oracle. Moreover, every computably enumerable set is Turing reducible to the diagonal halting set $K$, and neither $A$ nor $B$ can have its Turing degree, since that would make it compute the other set. HenceThis also gives many-one incomparability and answers the existence of intermediate computably enumerable Turing degrees.

A priority restraint protects finitely many oracle answers by requiring lower-priority enumerations to avoid positions below a numerical bound. Higher-priority action can cancel the priority restraint by initializing its owner. After the last such injury, the protected computation is permanent.