Assuming that the number of individuals in the universe is not finite, we have now succeeded not only in defining Peano's [Pg 24] 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 ," or simply "-hereditary," if, whenever the property belongs to a number , it also belongs to , i.e. to the number to which has the relation . And a number will be said to belong to the "posterity" of with respect to the relation if has every -hereditary property belonging to . These definitions can all be applied to any other relation just as well as to . Thus if is any relation whatever, we can lay down the following definitions:[8]— [8]These definitions, and the generalised theory of induction, are due to Frege, and were published so long ago as 1879 in his Begriffsschrift. In spite of the great value of this work, I was, I believe, the first person who ever read it—more than twenty years after its publication. A property is called "-hereditary" when, if it belongs to a term , and has the relation to , then it belongs to . A class is -hereditary when its defining property is -hereditary. A term is said to be an "-ancestor" of the term if has every -hereditary property that has, provided is a term which has the relation to something or to which something has the relation . (This is only to exclude trivial cases.) [Pg 25] The "-posterity" of is all the terms of which is an -ancestor. We have framed the above definitions so that if a term is the ancestor of anything it is its own ancestor and belongs to its own posterity. This is merely for convenience.