Editing Naive set theory (section)

===Cantor's theory===
Some believe that [[Georg Cantor]]'s set theory was not actually implicated in the set-theoretic paradoxes (see Frápolli 1991). One difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system. By 1899, Cantor was aware of some of the paradoxes following from unrestricted interpretation of his theory, for instance [[Cantor's paradox]]<ref name=Letter_to_Hilbert>Letter from Cantor to [[David Hilbert]] on September 26, 1897, {{harvnb|Meschkowski|Nilson|1991}} p. 388.</ref> and the [[Burali-Forti paradox]],<ref>Letter from Cantor to [[Richard Dedekind]] on August 3, 1899, {{harvnb|Meschkowski|Nilson|1991}} p. 408.</ref> and did not believe that they discredited his theory.<ref name=Letters_to_Dedekind>Letters from Cantor to [[Richard Dedekind]] on August 3, 1899 and on August 30, 1899, {{harvnb|Zermelo|1932}} p. 448 (System aller denkbaren Klassen) and {{harvnb|Meschkowski|Nilson|1991}} p. 407. (There is no set of all sets.)</ref> Cantor's paradox can actually be derived from the above (false) assumption—that any property {{math|''P''(''x'')}} may be used to form a set—using for {{math|''P''(''x'')}} "{{mvar|x}} is a [[cardinal number]]". Frege explicitly axiomatized a theory in which a formalized version of naive set theory can be interpreted, and it is ''this'' formal theory which [[Bertrand Russell]] actually addressed when he presented his paradox, not necessarily a theory Cantor{{--}}who, as mentioned, was aware of several paradoxes{{--}}presumably had in mind.