<P> Set theory began with the work of the logicians with the notion of "class" (modern "set") for example De Morgan (1847), Jevons (1880), Venn (1881), Frege (1879) and Peano (1889). It was given a push by Georg Cantor's attempt to define the infinite in set - theoretic treatment (1870--1890) and a subsequent discovery of an antinomy (contradiction, paradox) in this treatment (Cantor's paradox), by Russell's discovery (1902) of an antinomy in Frege's 1879 (Russell's paradox), by the discovery of more antinomies in the early 20th century (e.g., the 1897 Burali - Forti paradox and the 1905 Richard paradox), and by resistance to Russell's complex treatment of logic and dislike of his axiom of reducibility (1908, 1910--1913) that he proposed as a means to evade the antinomies . </P> <P> In 1902 Russell sent a letter to Frege pointing out that Frege's 1879 Begriffsschrift allowed a function to be an argument of itself: "On the other hand, it may also be that the argument is determinate and the function indeterminate ..." From this unconstrained situation Russell was able to form a paradox: </P> <Dl> <Dd> "You state...that a function, too, can act as the indeterminate element . This I formerly believed, but now this view seems doubtful to me because of the following contradiction . Let w be the predicate: to be a predicate that cannot be predicated of itself . Can w be predicated of itself?" </Dd> </Dl> <Dd> "You state...that a function, too, can act as the indeterminate element . This I formerly believed, but now this view seems doubtful to me because of the following contradiction . Let w be the predicate: to be a predicate that cannot be predicated of itself . Can w be predicated of itself?" </Dd>

When did the use of function notation y=f(x) first appear