Editing Cantor–Bernstein theorem (section)

{{short description|There are equally many countable order types and real numbers}}
{{For|the theorem that injections from A to B and from B to A imply a bijection between A and B|Schröder–Bernstein theorem}}
In [[set theory]] and [[order theory]], the '''Cantor–Bernstein theorem''' states that the [[cardinality]] of the second type class, the class of [[Countable set|countable]] [[order type]]s, equals the [[cardinality of the continuum]]. It was used by [[Felix Hausdorff]] and named by him after [[Georg Cantor]] and [[Felix Bernstein (mathematician)|Felix Bernstein]]. Cantor constructed a family of countable order types with the cardinality of the continuum, and in his 1901 inaugural dissertation Bernstein proved that such a family can have no higher cardinality.<ref name="plotkin">{{cite book|title=Hausdorff on Ordered Sets|volume=25|series=History of Mathematics|editor-first=J. M.|editor-last=Plotkin|publisher=American Mathematical Society|isbn=9780821890516|year=2005|page=3|url=https://books.google.com/books?id=M_skkA3r-QAC&pg=PA3}}.</ref>