Cantor's first set theory article: Difference between revisions

Revision as of 22:50, 6 June 2005

Revision as of 22:52, 7 June 2005

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The two monotone sequences ''a'' and ''b'' move toward each other. By the "gaplessness" of '''R''', some point ''c'' must lie between them. The claim is that ''c'' cannot be in the range of the sequence ''x'', and that is the contradiction. If ''c'' were in the range, then we would have ''c'' = ''x''_''i'' for some index ''i''. But then, when that index was reached in the process of defining ''a'' and ''b'', then ''c'' would have been added as the next member of one or the other of those two sequences, contrary to the assumption that it lies between their ranges.

===Real algebraic numbers and real transcendental numbers===

In the same paper, published in 1874, Cantor showed that the set of all real [[algebraic number]]s is countable, and inferred the existence of [[transcendental number]]s as a corollary. That corollary had earlier been proved by quite different methods by [[Joseph Liouville]].

===See also===

* [[Dedekind cut]]

==Reference==

* Georg Cantor, 1874, "Über eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen", ''Journal für die Reine und Angewandte Mathematik'', volume 77, pages 258-262.

[[Category:Set theory]]

[[de:Cantors erster Überabzählbarkeitsbeweis]]