Page:Cantortransfinite.djvu/128

cardinal number $\aleph _{0}$ and derived its properties that lie the most readily to hand, the question arises as to the higher cardinal numbers and how they proceed from $\aleph _{0}$. We shall show that the trans- finite cardinal numbers can be arranged according to their magnitude, and, in this order, form, like the finite numbers, a "well-ordered aggregate" in an extended sense of the words. Out of $\aleph _{0}$ proceeds, by a definite law, the next greater cardinal number $\aleph _{1}$, out of this by the same law the next greater $\aleph _{2}$ and so on. But even the unlimited sequence of cardinal numbers

$\aleph _{0},\aleph _{1},\aleph _{2},...,\aleph _{\nu },...$

does not exhaust the conception of transfinite cardinal number. We will prove the existence of a cardinal number which we denote by $\aleph _{\omega }$ and which shows itself to be the next greater to all the numbers $\aleph _{\nu }$; out of it proceeds in the same way as $\aleph _{1}$ out of $\aleph$ a next greater $\aleph _{\omega +1}$, and so on, without end.

[496] To every transfinite cardinal number ${\mathfrak {a}}$ there is a next greater proceeding out of it according to a unitary law, and also to every unlimitedly ascending well-ordered aggregate of transfinite cardinal numbers, $\{{\mathfrak {a}}\}$, there is a next greater proceeding out of that aggregate in a unitary way.

For the rigorous foundation of this matter, discovered in 1882 and exposed in the pamphlet Grundlagen einer allgemeinen Mannichfaltigkeitslehre (Leipzig, 1883) and in volume xxi of the