Alternatives to Zermelo ' S Assumption

1. The axiom of choice. The object of this paper is to consider the possibUity of setting up a logic in which the axiom of choice is false. The way of approach is through the second ordinal class, in connection with which there appear certain alternatives to the axiom of choice. But these alternatives have consequences not only with regard to the second ordinal class but also with regard to other classes, whose definitions do not involve the second ordinal class, in particular with regard to the continuum. And therefore it is possible to consider these alternatives as, in some sense, postulates of logic. In what follows we proceed, after certain introductory considerations, to state these postulates, to inquire into their character, and to derive as many as possible of their consequences. The axiom of choice, which is also known as Zermelo's assumption,f and, in a weakened form, as the multiplicative axiom,f is a postulate of logic which may be stated in the following way: Given any set X of classes which does not contain the null class, there exists a one-valued function, F, such that if x is any class of the set X then F(x) is a member of the class x. An equivalent statement is that there exists an assignment to every class x belonging to the set X of a unique element p such that p is contained in x. The important case is that in which the set X contains an infinite number of classes, because the assertion of the postulate is obviously capable of proof when the number of classes is finite. Accordingly a convenient, although not quite precise, characterization of the axiom of choice is obtained by saying that it is a postulate which justifies the employment of an infinite number of acts of arbitrary choice.