 The transfinite ordinals were introduced as new numbers in an important mathematico-philosophical paper of 1883, Grundlagen einer allgemeinen Mannigfaltigkeitslehre (notice that Cantor still uses Riemann’s term Mannigfaltigkeit or ‘manifold’ to denote sets). Cantor defined them by means of two “generating principles”: the first (1) yields the successor \(a+1\) for any given number \(a\), while the second (2) stipulates that there is a number \(b\) which follows immediately after any given sequence of numbers without a last element. Thus, after all the finite numbers comes, by (2), the first transfinite number, \(\omega\) (read: omega); and this is followed by \(\omega+1\), \(\omega+2\), …, \(\omega+\omega = \omega \cdot 2\), …, \(\omega \cdot n\), \(\omega\cdot n +1\), …, \(\omega^{2}\), \(\omega^{2}+1\), …, \(\omega^{\omega}\), … and so on and on. Whenever a sequence without last element appears, one can go on and, so to say, jump to a higher stage by (2).