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Cantor later formulated his second uncountability proof in 1877, known as '''[[Cantor's diagonal argument]]''', which proved the same thing but employed a method generally regarded as simpler and more elegant than the first.
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Cantor published the paper under what noted Cantor biographer [[Joseph W. Dauben]] <ref>{{cite journal|last=Dauben|first=Joseph W|title=Georg Cantor and the Battle for Transfinite Set Theory|journal= Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, CA)|pages=4|year=1993,2004}}</ref> calls a "very strange title", namely "On a Property of the Collection of All Real Algebraic Numbers" (German:''Über eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen'').
==The theorem==
Suppose a set '''R''' is
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The set of [[real numbers]] with its usual ordering is a typical example of such an ordered set '''R'''. The set of [[rational numbers]] (which ''is'' countable) has properties 1-3 but does not have property 4.
The proof is [[indirect proof|by contradiction]]. It begins by assuming '''R''' is countable and thus that some [[sequence]] ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ... has all of '''R''' as its range. Define two other sequences (''a''<sub>''n''</sub>) and (''b''<sub>''n''</sub>) as follows:
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