ABSTRACT
The series of natural numbers can all be defined if we know what we mean by the three terms “0,” “number,” and “successor.” Assuming that the number of individuals in the universe is not finite, we have now succeeded not only in defining Peano’s three primitive ideas, but in seeing how to prove his five primitive propositions, by means of primitive ideas and propositions belonging to logic. It follows that all pure mathematics, in so far as it is deducible from the theory of the natural numbers, is only a prolongation of logic. The process of mathematical induction, by means of which we defined the natural numbers, is capable of generalisation. The use of mathematical induction in demonstrations was, in the past, something of a mystery. Mathematical induction affords, more than anything else, the essential characteristic by which the finite is distinguished from the infinite.
