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[[File:Adolf Abraham Halevi Fraenkel.jpg|thumb|upright=0.93|alt=refer to caption|Abraham Fraenkel, between 1939 and 1949]]
As early as 1930, some mathematicians have attempted to correct this misconception of Cantor's work. In that year, the set theorist [[Abraham Fraenkel]] stated that Cantor's method is "… a method that incidentally, contrary to a widespread interpretation, is fundamentally constructive and not merely existential."<ref>{{harvnb|Fraenkel|1930|p=237}}. English translation: {{harvnb|Gray|1994|p=823}}.</ref> In 1972, [[Irving Kaplansky]] wrote: "It is often said that Cantor's proof is not 'constructive,' and so does not yield a tangible transcendental number. This remark is not justified. If we set up a definite listing of all algebraic numbers … and then apply the [[Cantor's diagonal argument|diagonal procedure]] …, we get a perfectly definite transcendental number (it could be computed to any number of decimal places)."<ref>{{harvnb|Kaplansky|1972|p=25}}.</ref>{{efn-ua|This proof is the same as Cantor's 1874 proof except for one modification: it uses his 1891 diagonal argument instead of his 1874 nested intervals argument to obtain a real.}}
Cantor's diagonal argument has often replaced his 1874 construction in expositions of his proof. The diagonal argument is constructive and produces a more efficient computer program than his 1874 construction. Using it, a computer program has been written that computes the digits of a transcendental number in [[polynomial time]]. The program that uses Cantor's 1874 construction requires at least [[sub-exponential time]].<ref>{{harvnb|Gray|1994|pp=821–824}}.</ref>{{efn-ua|The program using the diagonal method produces <math>n</math> digits in [[Big O notation#Use in computer science|<math>{\color{Blue}O}(n^2 \log^2 n \log \log n)</math>]] steps, while the program using the 1874 method requires at least <math>O(2^{\sqrt[3]{n}})</math> steps to produce <math>n</math> digits. ({{harvnb|Gray|1994|pp=822–823}}.)}}
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