What applies to exponentiation applies also to the product of two cardinals. We might define "" as the sum of the numbers of classes each having terms, but we prefer to define it as the number of ordered couples to be formed consisting of a member of followed by a member of , where has terms and has terms. This definition, also, is designed to evade the necessity of assuming the multiplicative axiom. [Pg 121] With our definitions, we can prove the usual formal laws of multiplication and exponentiation. But there is one thing we cannot prove: we cannot prove that a product is only zero when one of its factors is zero. We can prove this when the number of factors is finite, but not when it is infinite. In other words, we cannot prove that, given a class of classes none of which is null, there must be selectors from them; or that, given a class of mutually exclusive classes, there must be at least one class consisting of one term out of each of the given classes. These things cannot be proved; and although, at first sight, they seem obviously true, yet reflection brings gradually increasing doubt, until at last we become content to register the assumption and its consequences, as we register the axiom of parallels, without assuming that we can know whether it is true or false. The assumption, loosely worded, is that selectors and selections exist when we should expect them. There are many equivalent ways of stating it precisely. We may begin with the following:— "Given any class of mutually exclusive classes, of which none is null, there is at least one class which has exactly one term in common with each of the given classes." This proposition we will call the "multiplicative axiom."[24] We will first give various equivalent forms of the proposition, and then consider certain ways in which its truth or falsehood is of interest to mathematics. [24]Principia Mathematica, vol. I. * 88. Also vol. III. * 257-258. The multiplicative axiom is equivalent to the proposition that a product is only zero when at least one of its factors is zero; i.e. that, if any number of cardinal numbers be multiplied together, the result cannot be 0 unless one of the numbers concerned is 0. The multiplicative axiom is equivalent to the proposition that, if be any relation, and any class contained in the converse domain of , then there is at least one one-many relation implying and having for its converse domain.