Cantor's first set theory article: Difference between revisions

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The presentation of the non-constructive proof without mentioning Cantor's constructive proof appears in some books that were quite successful as measured by the length of time new editions or reprints appeared—for example: Oskar Perron's Irrationalzahlen (1921; 1960, 4th edition), [[Eric Temple Bell|Eric Temple Bell's]] ''[[Men of Mathematics]]'' (1937; still being reprinted), [[Godfrey Hardy]] and [[E. M. Wright|E. M. Wright's]] ''An Introduction to the [[Theory of numbers|Theory of Numbers]]'' (1938; 2008 6th edition), [[Garrett Birkhoff]] and [[Saunders Mac Lane|Saunders Mac Lane's]] ''A Survey of [[Modern Algebra]]'' (1941; 1997 5th edition), and [[Michael Spivak|Michael Spivak's]] ''[[Calculus]]'' (1967; 2008 4th edition).<ref>{{harvnb|Bell|1937|pp=568&ndash;569}}; {{harvnb|Hardy|Wright|1938|p=159}} (6th ed., pp. 205&ndash;206); {{harvnb|Birkhoff|Mac Lane|1941|p=392}}, (5th ed., pp. 436&ndash;437); {{harvnb|Spivak|1967|pp=369&ndash;370}} (4th ed., pp. 448&ndash;449).</ref>{{efn-ua|Starting with Hardy and Wright's book, these books are linked to Perron's book via their bibliographies: Perron's book is mentioned in the bibliography of Hardy and Wright's book, which in turn is mentioned in the bibliography of Birkhoff and Mac Lane's book and in the bibliography of Spivak's book. ({{harvnb|Hardy|Wright|1938|p=400}}; {{harvnb|Birkhoff|Mac Lane|1941|p=441}}; {{harvnb|Spivak|1967|p=515}}.)}} Since 2014, at least two books have appeared stating that Cantor's proof is constructive,<ref>{{harvnb|Dasgupta|2014|p=107}}; {{harvnb|Sheppard|2014|pp=131&ndash;132}}.</ref> and at least four have appeared stating that his proof does not construct any (or a single) transcendental.<ref>{{harvnb|Jarvis|2014|p=18}}; {{harvnb|Chowdhary|2015|p=19}}; {{harvnb|Stewart|2015|p=285}}; {{harvnb|Stewart|Tall|2015|p=333}}.</ref>

Asserting that Cantor gave a non-constructive argument without mentioning the constructive proof he published can lead to erroneous statements about the [[history of mathematics]]. In ''A Survey of Modern Algebra,'' Birkhoff and Mac Lane state: "Cantor's argument for this result [Not every real number is algebraic] was at first rejected by many mathematicians, since it did not exhibit any specific transcendental number." <ref>{{harvnb|Birkhoff|Mac Lane|1941|p=392}}, (5th ed., pp. 436&ndash;437).</ref> The proof that Cantor published produces transcendental numbers, and there appears to be no evidence that his argument was rejected. Even [[Leopold Kronecker]], who had strict views on what is acceptable in mathematics and who could have delayed publication of Cantor's article, did not delay it.<ref name=Gray828/> In fact, applying Cantor's construction to the sequence of real algebraic numbers produces a limiting process that Kronecker accepted—namely, it determines a number to any required degree of accuracy.{{efn-ua|Kronecker's opinion was: "Definitions must contain the means of reaching a decision in a finite number of steps, and existence proofs must be conducted so that the quantity in question can be calculated with any required degree of accuracy."<ref>{{harvnb|Burton|1995|p=595}}.</ref> So Kronecker would accept Cantor's argument as a valid existence proof, but he would not accept its conclusion that transcendental numbers exist. For Kronecker, they do not exist because their definition contains no means for deciding in a finite number of steps whether or not a given number is transcendental.<ref>{{harvnb|Dauben|1979|p=69}}.</ref> Cantor's 1874 construction calculates numbers to any required degree of accuracy because: Given a ''k'', an ''n'' can be computed such that {{nowrap|''b''_''n'' – ''a_n'' ≤ {{sfrac|1|''k''}}}} where (''a_n'',&nbsp;''b_n'') is the {{nowrap|''n''-th}} interval of Cantor's construction. An example of how to prove this is given in {{harvnb|Gray|1994|p=822}}. Cantor's diagonal argument provides an accuracy of 10^−''n'' after ''n'' real algebraic numbers have been calculated because each of these numbers generates one digit of the transcendental number.<ref>{{harvnb|Gray|1994|p=824}}.</ref>}}

== The influence of Weierstrass and Kronecker on Cantor's article ==