ABSTRACT
This chapter discusses an axiom which can be enunciated, but not proved, in terms of logic, and which is convenient, though not indispensable, in certain portions of mathematics. In defining the arithmetical operations, the only correct procedure is to construct an actual class having the required number of terms. 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 has been shown by Zermelo to be equivalent to the proposition that every class can be well-ordered, i.e. can be arranged in a series in which every sub-class has a first term.
