Cantor's first set theory article: Difference between revisions

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[[File:Georg Cantor3.jpg|thumb|alt=refer to caption|Georg Cantor, {{spaces|4|hair}}{{circa}} 1870]]

'''Cantor's first set theory article''' contains [[Georg Cantor]]'s first theorems of transfinite [[set theory]], which studies [[infinite set]]s and their properties. One of these theorems is his "revolutionary discovery" that the [[set (mathematics)|set]] of all [[real number]]s is [[uncountable set|uncountably]], rather than [[countable set|countably]], infinite.<ref>{{harvnb|Dauben|1993|p=4}}.</ref> This theorem is proved using '''Cantor's first uncountability proof''', which differs from the more familiar proof using his [[Cantor's diagonal argument|diagonal argument]]. The title of the article, "'''On a Property of the Collection of All Real Algebraic Numbers'''" ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set of real [[algebraic numbers]] is countable. Cantor's article was published in 1874. In 1879, he modified his uncountability proof by using the [[topological]] notion of a set being [[dense set|dense]] in an interval.

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Cantor's published proof and the reverse-order proof both use the theorem: Given a sequence of reals, a real can found that is not in the sequence. By applying this theorem to the sequence of real algebraic numbers, Cantor produced a transcendental number. He then proved that the reals are uncountable: Assume that there is a sequence containing all the reals. Applying the theorem to this sequence produces a real not in the sequence, contradicting the assumption that the sequence contains all the reals. Hence, the reals are uncountable.<ref name=Ewald840_841/> The reverse-order proof starts by first proving the reals are uncountable. It then proves that transcendental numbers exist: If there were no transcendental numbers, all the reals would be algebraic and hence countable, which contradicts what was just proved. This contradiction proves that transcendental numbers exist without constructing any.<ref name=Kanamori4/>

[[File:Oskar Perron.jpg|thumb|upright=0.93|alt=Oskar Perron reading a book while standing in front of a blackboard containing equations|Oskar Perron, circa{{spaces|4|hair}}c. 1948]]

The correspondence containing Cantor's non-constructive reasoning was published in 1937. By then, other mathematicians had rediscovered his non-constructive, reverse-order proof. As early as 1921, this proof was called "Cantor's proof" and criticized for not producing any transcendental numbers.<ref>{{harvnb|Gray|1994|pp=827&ndash;828}}.</ref> In that year, [[Oskar Perron]] gave the reverse-order proof and then stated: "… Cantor's proof for the existence of transcendental numbers has, along with its simplicity and elegance, the great disadvantage that it is only an existence proof; it does not enable us to actually specify even a single transcendental number."<ref>{{harvnb|Perron|1921|p=162}}</ref>{{efn-ua|By "Cantor's proof," Perron does not mean that it is a proof published by Cantor. Rather, he means that the proof only uses arguments that Cantor published. For example, to obtain a real not in a given sequence, Perron follows Cantor's 1874 proof except for one modification: he uses Cantor's 1891 diagonal argument instead of his 1874 nested intervals argument to obtain a real. Cantor never used his diagonal argument to reprove this theorem. In this case, both Cantor's proof and Perron's proof are constructive, so no misconception can arise here. Then, Perron modifies Cantor's proof of the existence of a transcendental by giving the reverse-order proof. This converts Cantor's 1874 constructive proof into a non-constructive proof which leads to the misconception about Cantor's work.}}

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==Dedekind's contributions to Cantor's article==

[[File:Dedekind.jpeg|thumb|upright=0.93|alt=refer to caption|Richard Dedekind, circa{{spaces|4|hair}}c. 1870]]

Since 1856, Dedekind had developed theories involving infinitely many infinite sets—for example: [[Ideal (ring theory)|ideal]]s, which he used in [[algebraic number theory]], and [[Dedekind cut]]s, which he used to construct the real numbers. This work enabled him to understand and contribute to Cantor's work.<ref>{{harvnb|Ferreirós|2007|pp=109&ndash;111, 172&ndash;174.}}</ref>