The Principles of Mathematics (1903)

§ 120

Having now clearly distinguished the finite from the infinite, we can devote ourselves to the consideration of finite numbers. It is customary, in the best treatises on the elements of Arithmetic[85], not to define number or particular finite numbers, but to begin with certain axioms or primitive propositions, from which all the ordinary results are shown to follow. This method makes Arithmetic into an independent study, instead of regarding it, as is done in the present work, as merely a development, without new axioms or indefinables, of a certain branch of general Logic. For this reason, the method in question seems to indicate a less degree of analysis than that adopted here. I shall nevertheless begin by an exposition of the more usual method, and then proceed to definitions and proofs of what are usually taken as indefinables and indemonstrables. For this purpose, I shall take Peano's exposition in the Formulaire[86], which is, so far as I know, the best from the point of view of accuracy and rigour. This exposition has the inestimable merit of showing that all Arithmetic can be developed from three fundamental notions (in addition to those of general Logic) and five fundamental propositions concerning these notions. It proves also that, if the three notions be regarded as determined by the five propositions, these five propositions are mutually independent. This is shown by finding, for each set of four out of the five propositions, an interpretation which renders the remaining proposition false. It therefore only remains, in order to connect Peano's theory with that here adopted, to give a definition of the three fundamental notions and a demonstration of the five fundamental propositions. When once this has been accomplished, we know with certainty that everything in the theory of finite integers follows.(§ 120 ¶ 1)

Peano's three indefinables are 0, finite integer[87], and successor of. It is assumed, as part of the idea of succession (though it would, I think, be better to state it as a separate axiom), that every number has one and only one successor. (By successor is meant, of course, immediate successor.) Peano's primitive propositions are then the following. 0 is a number. If a is a number, the successor of a is a number. If two numbers have the same successor, the two numbers are identical. 0 is not the successor of any number. If s be a class to which belongs 0 and also the successor of every number belonging to s, then every number belongs to s. The last of these propositions is the principle of mathematical induction.(§ 120 ¶ 2)

§ 120 n. 1. Except Frege's Grundgesetze der Arithmetik (Jena, 1893).

§ 120 n. 2. F. 1901, Part II and F. 1899, § 20 ff. F. 1901 differs from earlier editions in making number is a class a primitive proposition. I regard this as unnecessary, since it is implied by 0 is a number. I therefore follow the earlier editions.

§ 120 n. 3. Throughout the rest of this chapter, I shall use number as synonymous with finite integer.