Some Remarks on Set Theory
P. Erdös · Annals of Mathematics · 1943
Let there be given n ordinals a,, a2, , * , a,n. It is well known that every ordinal can be written uniquely as the sum of indecomposable ordinals. (An ordinal is said to be indecomposable if it is not the sum of two smaller ordinals.) Denote by +(a) the largest of these indecomposable ordinals belonging to a. ( (aj may have a coefficient c in the decomposition of a.) Put y = mini6, (4ai), and assume that there are k a's with o(ac) =,y. Denote these a's by a1, a2, * * *, ak. If in the sum ail+ai,+ +ain, ij, i2, * *, i,, a permutation of 1, 2, *, n, none of the ai, i < k appear at the end, they get absorbed in the following summands, and we get exactlyf(n-k) different sums. Assume next that exactly r of the ai's, r<k, appear at the end of ai,+ai2 + +aj. . Put