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Rh 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