Editing Dedekind-infinite set (section)

==History==
The term is named after the German mathematician [[Richard Dedekind]], who first explicitly introduced the definition. It is notable that this definition was the first definition of "infinite" that did not rely on the definition of the [[natural number]]s (unless one follows Poincaré and regards the notion of number as prior to even the notion of set). Although such a definition was known to [[Bernard Bolzano]], he was prevented from publishing his work in any but the most obscure journals by the terms of his political exile from the [[Charles University in Prague|University of Prague]] in 1819. Moreover, Bolzano's definition was more accurately a relation that held between two infinite sets, rather than a definition of an infinite set ''per se''.

For a long time, many mathematicians did not even entertain the thought that there might be a distinction between the notions of infinite set and Dedekind-infinite set. In fact, the distinction was not really realised until after [[Ernst Zermelo]] formulated the AC explicitly. The existence of infinite, Dedekind-finite sets was studied by [[Bertrand Russell]] and [[Alfred North Whitehead]] in 1912; these sets were at first called ''mediate cardinals'' or ''Dedekind cardinals''.

With the general acceptance of the axiom of choice among the mathematical community, these issues relating to infinite and Dedekind-infinite sets have become less central to most mathematicians. However, the study of Dedekind-infinite sets played an important role in the attempt to clarify the boundary between the finite and the infinite, and also an important role in the history of the AC.