EnumerationAn enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration (for example, whether the set must be finite, or whether the list is allowed to contain repetitions) depend on the discipline of study and the context of a given problem. Some sets can be enumerated by means of a natural ordering (such as 1, 2, 3, 4, ...
Cantor's first set theory articleCantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument.
Argument de la diagonale de Cantorvignette|Illustration de la diagonale de Cantor En mathématiques, l'argument de la diagonale, ou argument diagonal, fut inventé par le mathématicien allemand Georg Cantor et publié en 1891. Il permit à ce dernier de donner une deuxième démonstration de la non-dénombrabilité de l'ensemble des nombres réels, beaucoup plus simple, selon Cantor lui-même, que la première qu'il avait publiée en 1874, et qui utilisait des arguments d'analyse, en particulier le théorème des segments emboîtés.
Definable real numberInformally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, , can be defined as the unique positive solution to the equation , and it can be constructed with a compass and straightedge. Different choices of a formal language or its interpretation give rise to different notions of definability.
