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 extension of this result to those modern branches of mathematics which are not deducible from the theory of the natural numbers offers no difficulty of principle, as we have shown elsewhere.[7]
[7]For geometry, in so far as it is not purely analytical, see Principles of Mathematics, part VI.; for rational dynamics, ibid., part VII.
The process of mathematical induction, by means of which we defined the natural numbers, is capable of generalisation. We defined the natural numbers as the "posterity" of 0 with respect to the relation of a number to its immediate successor. If we call this relation
, any number
will have this relation to
. A property is "hereditary with respect to