bisections, the length of our bit is
of an inch; and this is finite whatever finite number
may be. The process of successive bisection does not lead to divisions whose ordinal number is infinite, since it is essentially a one-by-one process. Thus infinitesimals are not to be reached in this way. Confusions on such topics have had much to do with the difficulties which have been found in the discussion of infinity and continuity.
CHAPTER XII
SELECTIONS AND THE MULTIPLICATIVE AXIOM
IN this chapter we have to consider 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. It is convenient, in the sense that many interesting propositions, which it seems natural to suppose true, cannot be proved without its help; but it is not indispensable, because even without those propositions the subjects in which they occur still exist, though in a somewhat mutilated form.
Before enunciating the multiplicative axiom, we must first explain the theory of selections, and the definition of multiplication when the number of factors may be infinite.
In defining the arithmetical operations, the only correct procedure is to construct an actual class (or relation, in the case of relation-numbers) having the required number of terms. This sometimes demands a certain amount of ingenuity, but it is essential in order to prove the existence of the number defined. Take, as the simplest example, the case of addition. Suppose we are given a cardinal number