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There are computably enumerable sets $A,B$ with $A/{\le }_{T}B$ and $B/{\le }_{T}A$. A finite-injury priority construction alternates requirements preventing oracle programs for $B$ from computing $A$ and programs for $A$ from computing $B$. Each requirement uses a private witness, enumerates it when the opposing computation gives zero, and restrains the oracle segment preserving that disagreement. Verification inducts on priority. Both degrees are noncomputable and strictly below the halting degree, resolving Post problem.

The incomparable-degree formulation is stated in Steffen Lempp's notes, section 1.1.

The use of a convergent oracle computation is one more than its largest queried position, or zero if there are no queries. Preserving the oracle below this number preserves the computation. This finite dependence supplies the restraints in a finite-injury priority construction.

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.

A many-one reduction $A{\le }_{m}B$ is a total computable function $f$ with $x\in A⟺f(x)\in B$. A Turing reduction $A{\le }_{T}B$ is an oracle machine which decides $A$ with oracle $B$. It may make several adaptive queries and combine their answers. Every many-one reduction is a Turing reduction, by asking the single question $f(x)\in B$. The converse fails: for the diagonal halting set $K$, its complement is Turing-reducible to $K$ by reversing one oracle answer, but is not many-one reducible to $K$. Such a reduction would make the complement computably enumerable, contradicting undecidability of the halting problem.

The Friedberg–Muchnik theorem asserts that there are computably enumerable sets $A,B$ with incomparable Turing degrees:Here is a finite-injury priority construction proving it. Enumerate all Turing functionals ${\Phi }_e$ and put the requirements in the priority orderStart with $A_0=B_0=\varnothing$. Each requirement has a fresh witness, a waiting or acted status, and a restraint on the other set. A restraint $u$ forbids lower-priority strategies from enumerating numbers below $u$ into that oracle. Globally remember all previously chosen witnesses, including abandoned ones, and never choose any of them again.

For $R_e$, choose an unused witness $x$ outside $A_s\cup B_s$, larger than the stage and every current higher-priority restraint. Keep it out of $A$. Wait until the bounded simulation ${\Phi }_{e,s}^{B_s}(x)$ converges with value zero. If this happens, enumerate $x$ into $A$, put a restraint $u$ on $B$, where $u$ is one more than the largest oracle query in the observed computation, and mark the strategy acted. For $S_e$, interchange $A$ and $B$.

At stage $s$, consider the first $s+1$ requirements. Choose the highest-priority one needing a witness or waiting with an observed zero computation. Perform its indicated action, then initialize all lower-priority requirements: abandon their witnesses, clear their restraints, and reset their status. Do nothing if no requirement needs attention. Since witness selection respects higher restraints, and higher actions initialize every lower strategy, all protected computations are respected by lower-priority enumerations. Each stage is a finite effective calculation, so $A={\bigcup }_s{A}_s$ and $B={\bigcup }_s{B}_s$ are computably enumerable sets.

Verify by induction along the priority order that every requirement is initialized only finitely often and acts only finitely often. The highest requirement chooses one witness and can diagonalize at most once. After all higher requirements have finished acting, the next requirement is never again initialized; it obtains a final witness and can diagonalize at most once. This proves finite injury for every requirement and ensures that no requirement is permanently starved by higher attention.

Consider $R_e$ after its last initialization. If it acts, its witness belongs to $A$, while the protected computation with oracle $B$ remains zero: higher strategies no longer act and lower ones respect its restraint. Thus ${\Phi }_e^B(x)\ne {\chi }_A(x)$. If it never acts, its private witness remains outside $A$. Were ${\Phi }_e^B={\chi }_A$, the computation on that witness would converge to zero with a finite oracle use. Since the increasing sets $B_s$ eventually agree with $B$ on this finite initial segment, a sufficiently late bounded simulation would detect that computation and make $R_e$ act, a contradiction. The symmetric argument verifies every $S_e$.

Thus the two sets have incomparable Turing degrees. Both are noncomputable, since a computable set is Turing-reducible to every oracle. Both are also strictly below the degree of the halting problem: every computably enumerable set is Turing-reducible to $K$, while completeness of one would make the other reducible to it. This supplies the intermediate degrees sought in Post problem.

To prevent a Turing functional ${\Phi }_e^B$ from computing the indicator function of an enumerated set $A$, reserve a permanently unique fresh witness $x$. On observing a convergent computation with oracle use $u$, enumerate $x$ into $A$ if the answer is zero; otherwise keep it out. Restrain subsequent lower-priority enumerations into $B$ below $u$. Higher-priority actions may initialize this strategy and retire its witness, but witnesses are never reused. In a finite-injury priority construction, mathematical induction on priority shows that the final computation, if observed, remains correct as a computation from the final oracle and disagrees with $A$ at $x$. If no computation is ever observed after the final initialization, the final oracle computation diverges, since any convergent computation uses only finitely many eventually stable oracle bits.