Some Remarks on Set Theory

Let there be given n ordinals cti, alt • • • , an. 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 (cti), and assume that there are k a's with (cti) = y. Denote these a's by oti, cti, • • • , a*. If in the sum a»14-Q!»j4■ ■ • 4-a,„, *i, ii, • • • , in a permutation of 1, 2, • ■ • , n, none of the a,, i — k appear at the end, they get absorbed in the following summands, and we get exactly f(n — k) different sums. Assume next that exactly r of the a.'s, r = &, appear at the end of a^+a^ 4• • • 4-«,n. Put