Editing Georg Cantor (section)

====One-to-one correspondence====
{{Main|Bijection}}
[[File:Bijection.svg|thumb|A bijective function]]
Cantor's 1874 [[Crelle's Journal|Crelle]] paper was the first to invoke the notion of a [[Bijection|1-to-1 correspondence]], though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of the [[unit square]] and the points of a unit [[line segment]]. In an 1877 letter to Richard Dedekind, Cantor proved a far [[Mathematical jargon#stronger|stronger]] result: for any positive integer ''n'', there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an [[n-dimensional space|''n''-dimensional space]]. About this discovery Cantor wrote to Dedekind: "{{lang|fr|Je le vois, mais je ne le crois pas!}}" ("I see it, but I don't believe it!")<ref>{{Cite book |last=Wallace |first=David Foster |year=2003|title=Everything and More: A Compact History of Infinity|place=New York|publisher=W.&nbsp;W. Norton and Company|isbn=978-0-393-00338-3|page=[https://archive.org/details/everythingmore00davi/page/259 259]|url=https://archive.org/details/everythingmore00davi/page/259}}</ref> The result that he found so astonishing has implications for geometry and the notion of [[dimension]].

In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence and introduced the notion of "[[cardinality|power]]" (a term he took from [[Jakob Steiner]]) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined [[countable set]]s (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the [[natural number]]s, and proved that the rational numbers are denumerable. He also proved that ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup> has the same power as the [[real number]]s '''R''', as does a countably infinite [[Cartesian product|product]] of copies of '''R'''. While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about [[dimension]], stressing that his [[Map (mathematics)|mapping]] between the [[unit interval]] and the unit square was not a [[continuous function|continuous]] one.

This paper displeased Kronecker and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and [[Karl Weierstrass]] supported its publication.<ref>[[#Dauben1979|Dauben 1979]], pp. 69, 324 ''63n''. The paper had been submitted in July 1877. Dedekind supported it, but delayed its publication due to Kronecker's opposition. Weierstrass actively supported it.</ref> Nevertheless, Cantor never again submitted anything to Crelle.