Mathematical logic and foundations Computability theory Problem

Post's problem

Builds on Computably enumerable set Turing reduction Turing degree

Problem

Is there a computably enumerable set $A \subseteq \N$ that is not computable and is not Turing equivalent to the halting problem? In terms of Turing degrees: is there a computably enumerable degree $\mathbf{a}$ with $\mathbf{0} < \mathbf{a} < \mathbf{0}'$ ?

Remarks

Every computably enumerable set $A$ satisfies $A \le_T \emptyset'$ , so the question asks whether the c.e. degrees contain anything besides $\mathbf{0}$ and $\mathbf{0}'$ . Emil Post posed the question in 1944. The answer is yes. Friedberg and Muchnik answered it independently, in 1957 and 1956, by constructing two c.e. sets that are not Turing reducible to each other; see the Friedberg–Muchnik theorem. Their method of construction is the priority method. Post's own attempt was to find a structural property of c.e. sets, such as being simple, that forces the degree to lie strictly between $\mathbf{0}$ and $\mathbf{0}'$ .

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Tag	`1RVF7DWG8C`
URI	`m2://1RVF7DWG8C`
Kind	problem
Writing	W0 stub
MSC	03D25, 03D28
Status	This copy of the site has no build data. Run `m2 status 1RVF7DWG8C`.
About	Turing degree, Computably enumerable set
Uses	Computably enumerable set, Turing reduction, Turing degree
Changed	2 November 2026

Post's problem

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Remarks

Reading path

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