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
