The successor of the number of terms in the class
is the number of terms in the class consisting of a together with
, where
is any term not belonging to the class.
Certain niceties are required to make this definition perfect, but they need not concern us.[5] It will be remembered that we have already given (in Chapter II.) a logical definition of the number of terms in a class, namely, we defined it as the set of all classes that are similar to the given class.
[5]See Principia Mathematica, vol. II. * 110.
We have thus reduced Peano's three primitive ideas to ideas of logic: we have given definitions of them which make them definite, no longer capable of an infinity of different meanings, as they were when they were only determinate to the extent of obeying Peano's five axioms. We have removed them from the fundamental apparatus of terms that must be merely apprehended, and have thus increased the deductive articulation of mathematics.