Editing Dedekind-infinite set (section)

{{Short description|Set with an equinumerous proper subset}}
{{Redirect|Dedekind finite|the term from ring theory|Dedekind-finite ring}}
In [[mathematics]], a set ''A'' is '''Dedekind-infinite''' (named after the German mathematician [[Richard Dedekind]]) if some proper [[subset]] ''B'' of ''A'' is [[equinumerous]] to ''A''. Explicitly, this means that there exists a [[bijective function]] from ''A'' onto some proper subset ''B'' of ''A''. A set is '''Dedekind-finite''' if it is not Dedekind-infinite (i.e., no such bijection exists). Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the [[natural number]]s.<ref name="moore">{{cite book |title=Zermelo's Axiom of Choice: Its Origins, Development & Influence |last=Moore |first=Gregory H. |year=2013 |orig-year=unabridged republication of the work originally published in 1982 as Volume 8 in the series "Studies in the History of Mathematics and Physical Sciences" by Springer-Verlag, New York |publisher=Dover Publications |isbn=978-0-486-48841-7}}</ref>

A simple example is <math>\mathbb{N}</math>, the set of [[natural number]]s. From [[Galileo's paradox]], there exists a bijection that maps every natural number ''n'' to its [[square number|square]] ''n''<sup>2</sup>. Since the set of squares is a proper subset of <math>\mathbb{N}</math>, <math>\mathbb{N}</math> is Dedekind-infinite. <!--This is just one example of the many bijective functions that exist for <math>\mathbb{N}</math>.-->

Until the [[foundational crisis of mathematics]] showed the need for a more careful treatment of set theory, most mathematicians [[tacit assumption|assumed]] that a set is [[infinite set|infinite]] [[if and only if]] it is Dedekind-infinite. In the early twentieth century, [[Zermelo–Fraenkel set theory]], today the most commonly used form of [[axiomatic set theory]], was proposed as an [[axiomatic system]] to formulate a [[theory of sets]] free of paradoxes such as [[Russell's paradox]]. Using the axioms of Zermelo–Fraenkel set theory with the originally highly controversial [[axiom of choice]] included ('''ZFC''') one can show that a set is Dedekind-finite if and only if it is [[finite set|finite]] in the usual sense. However, there exists a model of Zermelo–Fraenkel set theory without the axiom of choice ('''ZF''') in which there exists an infinite, Dedekind-finite set, showing that the axioms of '''ZF''' are not strong enough to prove that every set that is Dedekind-finite is finite.<ref name="herrlich">{{cite book |title=Axiom of Choice |last=Herrlich | first=Horst |year=2006 |publisher=Springer-Verlag |series=Lecture Notes in Mathematics 1876 |isbn=978-3540309895}}</ref><ref name="moore"/> There are [[finite set#Necessary and sufficient conditions for finiteness|definitions of finiteness and infiniteness of sets]] besides the one given by Dedekind that do not depend on the axiom of choice.

A vaguely related notion is that of a [[Dedekind-finite ring]].