Many other forms of the axiom might be given, but the above are the most important of the forms known at present. As to the truth or falsehood of the axiom in any of its forms, nothing is known at present. The propositions that depend upon the axiom, without being known to be equivalent to it, are numerous and important. Take first the connection of addition and multiplication. We naturally think that the sum of mutually exclusive classes, each having terms, must have terms. When is finite, this can be proved. But when is infinite, it cannot be proved without the multiplicative axiom, except where, owing to some special circumstance, the existence of certain selectors can be proved. The way the multiplicative axiom enters in is as follows: Suppose we have two sets of mutually exclusive classes, each having terms, and we wish to prove that the sum of one set has as many terms as the sum of the other. In order to prove this, we must establish a one-one relation. Now, since there are in each case classes, there is some one-one relation between the two sets of classes; but what we want is a one-one relation between their terms. Let us consider some one-one relation between the classes. Then if and are the two sets of classes, and is some member of , there will be a member of which will be the correlate of with respect to . Now and each have terms, and are therefore similar. There are, accordingly, one-one correlations of and . The trouble is that there are so many. In order to obtain a one-one correlation of the sum of with the sum of , we have to pick out one selection from a set of classes [Pg 124] of correlators, one class of the set being all the one-one correlators of with . If and are infinite, we cannot in general know that such a selection exists, unless we can know that the multiplicative axiom is true. Hence we cannot establish the usual kind of connection between addition and multiplication.