5. The Reality is such a self-represented and infinite system. And herein lies the basis of its very union, within itself of the one and the many. For the one purpose of self-representa tion demands an infinite multiplicity to express it; while no multiplicity is reducible to unity except through processes involving self-repre sentation.
6. Nevertheless, the Real is exclusive as well as inclusive. On the side of its thought the Absolute does conceive a barely possible infinity, other than the real infinity, a possible world,. whose characters, as universal characters, are present to the Absolute, and are known by virtue of the fact the Absolute thinks.
The Royceian theory of the infinite is based on the analogy of cardinal infinitude, and supposes that there is such a thing as a plete universe. Certain paradoxes discovered by Russell and Burali-Forti tell very strongly against the existence of a complete unity bracing all lesser unities. Royce's work sesses value rather as an account of the tential infinity of a universe capable of inite enlargement than as a description of- a given, complete infinite. Furthermore, it is
clear that systems of much less extent than the Royceian infinite maypossess the self-reflecting property of cardinal infinity. (See ASSEMBLAGES, GENERAL THEORY or). Consult Bosanquet, B., .(2d ed., Oxford 1911); Bobzano, des Unendlichen) (Leipzig 1851) - Bradley, 'Appearance and Reality) (2d ed., New York 1902) ; Cohn, J., des Unendlichkeitsproblem in abendlindischen Denken bis Kant) (Leipzig 1896) • Coutural, 'De l'infini mathematique) (Paris 1896); Geissler, Grundsitze und das Wesen des Unendlichen in der Mathematik und Philosophic' (Leipzig 1902) • Hegel, 'Wissenschaft der Logik) lin 1841) i Russell, B. A. W., 'Principles of Mathematics) (Cambridge 1903) ; Royce, J., The World and the Individual) (New York 1900-01) ; Whitehead, A. N., and Russell, B. A. W. Matheznatica) (Cambridge 1910). NORBERT WIENER, Editorial Staff of The Americana.