On Cardinal Numbers. By A. N. Whitehead, Fellow of Trinity College, Cambridge, England. Preface. The following memoir is concerned solely with the theory of Cardinal num- bers, and does not touch upon the theory of ordered aggregates or of ordinal types which form the other parts of the general theory of aggregates (mengenlehre).* Sections I and II are preliminary, to explain the notation and methods of reasoning. Incidentally they may be of independent service as explaining the elements of Peano's developments of mathematical logicf (Section I) and of Russell's symbolism for the Logic of Relations J (Section II). I believe that these two methods are almost indispensable for the development of the theory of Cardinal numbers. The abstract nature of the subject makes ordinary language totally ineffective, only gaining precision by verbosity, and imagination is very misleading, since it presents to us special aggregates which are denumerable or of the power of the continuum. Thus we are thrown back onto a strict logical deduction by a symbolical method. I believe that the invention of the Peano and Russell symbolism, used here, forms an epoch in mathematical reasoning.

- Due almost entirely to G. Cantor. Almost all that is known respecting Cardinal Numbers, apart

from their relations to the other parts of the theory, is given in his memoir in Math. Annal., Bd. XL VI, 1895 : this and its sequel on Ordinal Types, loo. cit. Bd. XLIX, can be obtained as one pamphlet in a French translation, translated by F. Marotte, Paris, A. Hermann, 1899. For a general sketch of the whole theory, with bibliography, cf. " Die Entwickelung der Lehre von den Punktmannigfaltigkeiten " : Bericht von A. Schonflies,"Yahresbericht der Deutschen Mathematiker- Vereinigung,' ' 1900. For a devel- opment of the theory of Ordinal Types in respect to condensed series, by the methods of the present memoir, cf. Bertrand Russell, Revue de Mathematiques, vol. VII, Turin, 1901, where the methods of reasoning used here were first elaborated. t For references to Peano's articles, cf. my memoir on the " Algebra of Symbolic Logic," in this Journal, vol. XXIII. % Cf. Russell, loc. cit. 48