Título: Georg Cantor
Seção: One-to-one correspondence
Background: Georg Ferdinand Ludwig Philipp Cantor ( KAN-tor; German: ['geoRk 'feRdinant 'lu:tvIc 'fIlIp 'kantoR]; March 3 [O.S. February 19] 1845 - January 6, 1918) was a German mathematician. He invented set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers.

Conteúdo: Cantor's 1874 Crelle paper was the first to invoke the notion of a 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 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. About this discovery Cantor wrote to Dedekind: "Je le vois, mais je ne le crois pas!" ("I see it, but I don't believe it!") 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 "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 sets (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the natural numbers, and proved that the rational numbers are denumerable. He also proved that n-dimensional Euclidean space Rn has the same power as the real numbers R, as does a countably infinite 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 hi

user: What is one-to-one correspondence?

assistant: 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.

user: What year was he doing his work on this?

assistant: In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence

user: What is Crelle's Journal?

assistant: CANNOTANSWER

user: How was his paper accepted?

assistant: This paper displeased Kronecker, and Cantor wanted to withdraw it;

user: Why did it displease Kronecker?

assistant: CANNOTANSWER

user: Did he work with any other collegues during this time?

assistant: In an 1877 letter to Richard Dedekind, Cantor proved a far stronger result: for any positive integer n, there exists a 1-to-1 correspondence between the points

user: How did Dedekind accept this new result?

assistant: Cantor wrote to Dedekind: "Je le vois, mais je ne le crois pas!" ("I see it, but I don't believe it!")

user: What else is interesting in this article?

assistant: Cantor also discussed his thinking about dimension, stressing that his mapping between the unit interval and the unit square was not a continuous one.