| dbo:abstract |
Cantors erster Überabzählbarkeitsbeweis ist Georg Cantors erster Beweis, dass die reellen Zahlen eine überabzählbare Menge bilden. Er kommt ohne das Dezimalsystem oder irgendein anderes Zahlensystem aus. Die Behauptung und der erste Beweis wurden von Cantor im Dezember 1873 entdeckt, und 1874 in Crelles Journal (Journal für die Reine und Angewandte Mathematik, Bd. 77, 1874) veröffentlicht. Viel bekannter wurde sein 1877 gefundener zweiter Beweis dafür, Cantors zweites Diagonalargument. (de) Cantor'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. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers" ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set of real algebraic numbers is countable. Cantor's article was published in 1874. In 1879, he modified his uncountability proof by using the topological notion of a set being dense in an interval. Cantor's article also contains a proof of the existence of transcendental numbers. Both constructive and non-constructive proofs have been presented as "Cantor's proof." The popularity of presenting a non-constructive proof has led to a misconception that Cantor's arguments are non-constructive. Since the proof that Cantor published either constructs transcendental numbers or does not, an analysis of his article can determine whether or not this proof is constructive. Cantor's correspondence with Richard Dedekind shows the development of his ideas and reveals that he had a choice between two proofs: a non-constructive proof that uses the uncountability of the real numbers and a constructive proof that does not use uncountability. Historians of mathematics have examined Cantor's article and the circumstances in which it was written. For example, they have discovered that Cantor was advised to leave out his uncountability theorem in the article he submitted — he added it during proofreading. They have traced this and other facts about the article to the influence of Karl Weierstrass and Leopold Kronecker. Historians have also studied Dedekind's contributions to the article, including his contributions to the theorem on the countability of the real algebraic numbers. In addition, they have recognized the role played by the uncountability theorem and the concept of countability in the development of set theory, measure theory, and the Lebesgue integral. (en) Cantors eerste overaftelbaarheidsbewijs toont aan dat de verzameling van alle reële getallen overaftelbaar is. Dit bewijs verschilt van het meer bekende bewijs, waarin Cantor zijn diagonaalargument gebruikt. Het eerste overaftelbaarheidsbewijs van Cantor werd in 1874 gepubliceerd, in een artikel dat ook een bewijs bevat dat de verzameling van de reële algebraïsche getallen aftelbaar is en een bewijs van het bestaan van transcendente getallen. (nl) |
| rdfs:comment |
Cantors erster Überabzählbarkeitsbeweis ist Georg Cantors erster Beweis, dass die reellen Zahlen eine überabzählbare Menge bilden. Er kommt ohne das Dezimalsystem oder irgendein anderes Zahlensystem aus. Die Behauptung und der erste Beweis wurden von Cantor im Dezember 1873 entdeckt, und 1874 in Crelles Journal (Journal für die Reine und Angewandte Mathematik, Bd. 77, 1874) veröffentlicht. Viel bekannter wurde sein 1877 gefundener zweiter Beweis dafür, Cantors zweites Diagonalargument. (de) Cantors eerste overaftelbaarheidsbewijs toont aan dat de verzameling van alle reële getallen overaftelbaar is. Dit bewijs verschilt van het meer bekende bewijs, waarin Cantor zijn diagonaalargument gebruikt. Het eerste overaftelbaarheidsbewijs van Cantor werd in 1874 gepubliceerd, in een artikel dat ook een bewijs bevat dat de verzameling van de reële algebraïsche getallen aftelbaar is en een bewijs van het bestaan van transcendente getallen. (nl) Cantor'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. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers" ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set of real algebraic numbers is countable. Cantor's article was published in 1874. In 1879, he modified his uncountability proof by using the topological notion of a set be (en) |
