Cantor's first set theory article: Difference between revisions

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[[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, {{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 thishis 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.}}

[[File:Adolf Abraham Halevi Fraenkel.jpg|thumb|upright=0.93|alt=refer to caption|Abraham Fraenkel, between 1939 and 1949]]