 For a proper understanding of the incompleteness and undecidability results, it is vital to understand the difference between the two key notions regarding sets. First, there may be a mechanical method which decides whether any given number belongs to the set at issue or not (in which case the set is called “decidable” or “recursive”), and, second, there may be a mechanical method which generates or lists the elements of the set, number by number. In the latter case, the set is called “recursively enumerable” (r.e.), that is to say, it can be effectively generated, or it is “semi-decidable.” It is a fundamental result of the theory of computability (or “the theory of recursive functions”) that there are semi-decidable sets, sets which can be effectively generated (i.e., are recursively enumerable), but are not decidable (i.e., not recursive). In fact, this is, in the very abstract level, the essence of the first incompleteness theorem. However, if both a set and its complement are recursively enumerable, the set is recursive, i.e., decidable.