It remains to define "successor." Given any number , let be a class which has members, and let be a term which is not a member of . Then the class consisting of with added on will have members. Thus we have the following definition:— 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 [Pg 23] 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. As regards the five primitive propositions, we have already succeeded in making two of them demonstrable by our definition of "natural number." How stands it with the remaining three? It is very easy to prove that 0 is not the successor of any number, and that the successor of any number is a number. But there is a difficulty about the remaining primitive proposition, namely, "no two numbers have the same successor." The difficulty does not arise unless the total number of individuals in the universe is finite; for given two numbers and , neither of which is the total number of individuals in the universe, it is easy to prove that we cannot have unless we have . But let us suppose that the total number of individuals in the universe were (say) 10; then there would be no class of 11 individuals, and the number 11 would be the null-class. So would the number 12. Thus we should have 11 = 12; therefore the successor of 10 would be the same as the successor of 11, although 10 would not be the same as 11. Thus we should have two different numbers with the same successor. This failure of the third axiom cannot arise, however, if the number of individuals in the world is not finite. We shall return to this topic at a later stage.[6] [6]See Chapter XIII.