 The introduction of these new numbers seemed like idle speculation to most of his contemporaries, but for Cantor they served two very important functions. To this end, he classified the transfinite ordinals as follows: the “first number class” consisted of the finite ordinals, the set \(\mathbf{N}\) of natural numbers; the “second number class” was formed by ω and all numbers following it (including \(\omega^{\omega}\), and many more) that have only a denumerable set of predecessors. This crucial condition was suggested by the problem of proving the Cantor-Bendixson theorem (see Ferreirós 1995). On that basis, Cantor could establish the results that the cardinality of the “second number class” is greater than that of \(\mathbf{N}\); and that no intermediate cardinality exists. Thus, if you write \(\textit{card}(\mathbf{N}) = \aleph_{0}\) (read: aleph zero), his theorems justified calling the cardinality of the “second number class” \(\aleph_{1}\).