This fact has various curious consequences. To begin with, we know that . It is commonly inferred from this that the sum of classes each having members must itself have members, but this inference is fallacious, since we do not know that the number of terms in such a sum is , nor consequently that it is . This has a bearing upon the theory of transfinite ordinals. It is easy to prove that an ordinal which has predecessors must be one of what Cantor calls the "second class," i.e. such that a series having this ordinal number will have terms in its field. It is also easy to see that, if we take any progression of ordinals of the second class, the predecessors of their limit form at most the sum of classes each having terms. It is inferred thence—fallaciously, unless the multiplicative axiom is true—that the predecessors of the limit are in number, and therefore that the limit is a number of the "second class." That is to say, it is supposed to be proved that any progression of ordinals of the second class has a limit which is again an ordinal of the second class. This proposition, with the corollary that (the smallest ordinal of the third class) is not the limit of any progression, is involved in most of the recognised theory of ordinals of the second class. In view of the way in which the multiplicative axiom is involved, the proposition and its corollary cannot be regarded as proved. They may be true, or they may not. All that can be said at present is that we do not know. Thus the greater part of the theory of ordinals of the second class must be regarded as unproved.