The Principles of Mathematics (1903)

§ 123

In order to bring out more clearly the difference between Peano's procedure and mine, I shall here repeat the definition of the class satisfying his five primitive propositions, the definition of finite number, and the proof, in the case of finite numbers, of his five primitive propositions.(§ 123 ¶ 1)

The class of classes satisfying his axioms is the same as the class of classes whose cardinal number is a0, i.e. the class of classes, according to my theory, which is a0. It is most simply defined as follows: a0 is the class of classes u each of which is the domain of some one-one relation R (the relation of a term to its successor) which is such that there is at least one term which succeeds no other term, every term which succeeds has a successor, and u is contained in any class s which contains a term of u having no predecessors, and also contains the successor of every term of u which belongs to s. This definition includes Peano's five primitive propositions and no more. Thus of every such class all the usual propositions of in the arithmetic of finite numbers can be proved: addition, multiplication, fractions, etc. can be defined, and the whole of analysis can be developed, in so far as complex numbers are not involved. But in this whole development, the meaning of the entities and relations which occur is to a certain degree indeterminate, since the entities and relations with which we start are variable members of a certain class. Moreover, in this whole development, nothing shows that there are such classes as the definition speaks of.(§ 123 ¶ 2)

In the logical theory of cardinals, we start from the opposite end. We first define a certain class of entities, and then show that this class of entities belongs to the class a0 above defined. This is done as follows. (1) 0 is the class of classes whose only member is the null-class. (2) A number is the class of all classes similar to any one of themselves. (3) 1 is the class of all classes which are not null and are such that, if x and y belong to the class, then x and y are identical. (4) Having shown that if two classes be similar, and a class of one term be added to each, the sums are similar, we define that if n be a number, n+1 is the number resulting from adding a unit to a class of n terms. (5) Finite numbers are those belonging to every class s to which belongs 0, and to which n+1 belongs if n belongs. This completes the definition of finite numbers. We then have, as regards the five propositions which Peano assumes: (1) 0 is a number. (2) Meaning n+1 by the successor of n, if n be a number, then n+1 is a number. (3) If n+1=m+1, then n=m. (4) If n be any number, n+1 is different from 0. (5) If s be a class, and 0 belongs to this class, and if when n belongs to it, n+1 belongs to it, then all finite numbers belong to it. Thus all the five essential properties are satisfied by the class of finite numbers as above defined. Hence the class of classes a0 has members, and the class finite number is one definite member of a0. There is, therefore, from the mathematical standpoint, no need whatever of new indefinables or indemonstrables in the whole of Arithmetic and Analysis.(§ 123 ¶ 3)