Computably enumerable set

A set is computably enumerable if an algorithm can list all its members, possibly repeating elements, without necessarily signaling when the list is complete.

Enumerator: An algorithm that outputs the members of a set, possibly with repetitions and without stopping. Its outputs provide the defining listing of a computably enumerable set.
Turing machine: An abstract model of computation that manipulates symbols according to finite rules. Its computations provide a precise model for algorithms that enumerate sets.
Rice's theorem: A theorem stating that every nontrivial semantic property of programs is undecidable. It shows why many questions about enumerated program behaviors cannot be decided.
Co-computably enumerable set: A set whose complement is computably enumerable. A set can be enumerable while its complement remains impossible to enumerate.
Semi-decidable set: A set whose members can be recognized by an algorithm that halts on members but may run forever otherwise. Running its recognizer on successive inputs yields an enumeration of the set.
Algorithm: A finite, precise procedure for carrying out a computation. An effective listing must be produced by such a procedure.
Recursively enumerable language: A formal language whose strings can be generated by an algorithm. It is the language-theoretic counterpart of a computably enumerable set.
Computable set: A set whose membership can be decided by an algorithm that always halts. Every computable set is enumerable, but some enumerable sets are not computable.
Halting problem: The decision problem of determining whether a program eventually stops on a given input. Its positive instances form a computably enumerable set, but its negative instances do not.
Formal language: A set of finite strings over an alphabet. Formal languages are commonly studied through whether their strings can be enumerated.

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Hilbert's tenth problemNarrower topic: Matiyasevich's theorem connects these sets to integer solutions of polynomial equations.