Cantor's first ten papers were on number theory, his thesis topic. At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to analysis. Heine proposed that Cantor solve an open Problem that had eluded Peter Gustav Lejeune Dirichlet, Rudolf Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series. Cantor solved this difficult Problem in 1869. It was while working on this Problem that he discovered transfinite ordinals, which occurred as indices *n* in the *n*th derived set *S*_{n} of a set *S* of zeros of a trigonometric series. Given a trigonometric series f(x) with *S* as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had *S*_{1} as its set of zeros, where *S*_{1} is the set of limit points of *S*. If *S*_{k+1} is the set of limit points of *S*_{k}, then he could construct a trigonometric series whose zeros are *S*_{k+1}. Because the sets *S*_{k} were closed, they contained their Limit points, and the intersection of the infinite decreasing sequence of sets *S*, *S*_{1}, *S*_{2}, *S*_{3},... formed a limit set, which we would now call *S*_{ω}, and then he noticed that *S*_{ω} would also have to have a set of limit points *S*_{ω+1}, and so on. He had examples that went on forever, and so here was a naturally occurring infinite sequence of infinite numbers *ω*, *ω* + 1, *ω* + 2, ...