 After the second number class comes a “third number class” (all transfinite ordinals whose set of predecessors has cardinality \(\aleph_{1}\)); the cardinality of this new number class can be proved to be \(\aleph_{2}\). And so on. The first function of the transfinite ordinals was, thus, to establish a well-defined scale of increasing transfinite cardinalities. (The aleph notation used above was introduced by Cantor only in 1895.) This made it possible to formulate much more precisely the problem of the continuum; Cantor’s conjecture became the hypothesis that \(\textit{card}(\mathbf{R}) = \aleph_{1}\). Furthermore, relying on the transfinite ordinals, Cantor was able to prove the Cantor-Bendixson theorem, rounding out the results on point sets that he had been elaborating during these crucial years. The Cantor-Bendixson theorem states: closed sets of \(\mathbf{R}^n\) (generalizable to Polish spaces) have the perfect set property, so that any closed set \(S\) in \(\mathbf{R}^n\) can be written uniquely as the disjoint union of a perfect set \(P\) and a countable set \(R\). Moreover, \(P\) is \(S^α\) for α countable ordinal.