ABSTRACT
It may happen that JT contains a member xx and an infinite sub-class rit not including xv such that the p-combinations consisting of xx together with any p — 1 members of r i f all belong to Cv If so, rt may similarly contain a member x2 and an infinite sub-class' rz, not including x2, such that all the p-combinations consisting of x2 together with p — 1 members of r2, belong to Cv And, again, r2 may contain an x3 and a jT3 with similar properties, and so on indefinitely. We thus have two possibilities: either we can select in this way two infinite sequences of members of P (xv x2, . . ., xnt . . . ) , and of infinite sub-classes of J1 ( r v JH2, . . JT»* • • •)> in which xn is always a member of rn-v and Fn a sub-class of JTn-i not including xn> such that all the ^-combinations consisting of xn together with p — 1 members of rn, belong to C1; or else the process of selection will fail at a certain stage, say the n-th, because r n . 1 (or if n = 1, JT itself) will contain no member xn and infinite sub-class jTn not including xn such that all the p-combinations consisting of xn together with p - 1 members of rn belong to Cv Let us take these possibilities in turn.
