User:RJGray/Cantor draft2
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Cantor's first uncountability proof demonstrates that the set of all real numbers is uncountably, rather than countably, infinite. This proof differs from the more familiar proof that uses his diagonal argument. Georg Cantor's first proof was published in 1874, in an article that also contains a proof that the set of real algebraic numbers is countable, and a proof of the existence of transcendental numbers.[1]
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/6/67/Georg_Cantor3.jpg/220px-Georg_Cantor3.jpg)
As early as 1930, some mathematicians have disagreed on whether Cantor's proof of the existence of transcendental numbers is constructive or non-constructive.[2] Books as recent as 2014 and 2015 indicate that this disagreement has not been resolved.[3] A careful study of Cantor's article will determine the nature of his proof. Cantor's correspondence shows the development of his ideas and reveals that he had a choice between two proofs: one uses the uncountability of the real numbers; the other does not. These proofs play an important role in the disagreement about his proof.
The title of Cantor's article is "On a Property of the Collection of All Real Algebraic Numbers." Historians of mathematics have studied the reasons why Cantor's article emphasizes the countability of the real algebraic numbers rather than the uncountability of the real numbers. They have discovered interesting facts about the article—for example, Cantor left out his uncountability theorem in the article he submitted, then added it during proofreading.[4] They have also studied Richard Dedekind's contributions to the article and the article's legacy.