Editing Countable set (section)

==History==
In 1874, in [[Georg Cantor's first set theory article|his first set theory article]], Cantor proved that the set of [[real number]]s is uncountable, thus showing that not all infinite sets are countable.<ref>{{citation|title=Roads to Infinity: The Mathematics of Truth and Proof|first=John C.|last=Stillwell|author-link=John Stillwell|publisher=CRC Press|year=2010| isbn=9781439865507|page=10|url=https://books.google.com/books?id=XvPRBQAAQBAJ&pg=PA10|quote=Cantor's discovery of uncountable sets in 1874 was one of the most unexpected events in the history of mathematics. Before 1874, infinity was not even considered a legitimate mathematical subject by most people, so the need to distinguish between countable and uncountable infinities could not have been imagined.}}</ref> In 1878, he used one-to-one correspondences to define and compare cardinalities.<ref>Cantor 1878, p.&nbsp;242.</ref> In 1883, he extended the natural numbers with his infinite [[ordinal number|ordinals]], and used sets of ordinals to produce an infinity of sets having different infinite cardinalities.<ref>Ferreirós 2007, pp. 268, 272&ndash;273.</ref>