Editing Computably enumerable set (section)

== Remarks ==
According to the [[Church–Turing thesis]], any effectively calculable function is calculable by a [[Turing machine]], and thus a set ''S'' is computably enumerable if and only if there is some [[algorithm]] which yields an enumeration of ''S''.  This cannot be taken as a formal definition, however, because the Church–Turing thesis is an informal conjecture rather than a formal axiom.

The definition of a computably enumerable set as the ''domain'' of a partial function, rather than the ''range'' of a total computable function, is common in contemporary texts.  This choice is motivated by the fact that in generalized recursion theories, such as [[Alpha recursion theory|α-recursion theory]], the definition corresponding to domains has been found to be more natural.  Other texts use the definition in terms of enumerations, which is equivalent for computably enumerable sets.