The Principles of Mathematics (1903)

Chapter XI. Definition of Cardinal Numbers

Table of Contents

  1. § 107. Plan of Part II
  2. § 108. Mathematical meaning of definition
  3. § 109. Definitions of numbers by abstraction
  4. § 110. Objections to this definition
  5. § 111. Nominal definition of numbers

§ 107 n. 1. Cantor has shown that it is necessary to separate the study of Cardinal and Ordinal numbers, which are distinct entities, of which the former are simpler, but of which both are essential to ordinary Mathematics. On Ordinal numbers, cf. Chaps. XXIX, XXXVIII, infra.

§ 108 n. 1. See Peano, F. 1901, p. 6 ff. and Padoa, Théorie Algébrique des Nombres Entiers, Congrès, Vol. III, p. 314 ff.

§ 108 n. 2. Cf. Burali-Forti, Sur les différentes définitions du nombre réel, Congrès, III, p. 294 ff.

§ 109 n. 1. Cf. Peano, F. 1901, § 32, ·0, Note.

§ 110 n. 1. On the necessity of this condition, cf. Padoa, loc. cit., p. 324. Padoa appears not to perceive, however, that all definitions define the single individual of a class; when what is defined is a class, this must be the only term of some class of classes.