B. RUSSELL ON THE INFINITY

The idea of infinity without which themathematics is impossible, is entered into system of Principia Mathematic by means of an infinityaxiom. The infinity axiom is needed in PM because some features of definition of natural numberconcept and axiomatization of arithmetic by J. Peano. However there is a question. Is it possible tointroduce the idea of infinity on other bases? There are such attempts. Purely analytically the idea ofinfinity can be introduced on the basis of the definition of the natural number concept offered by G.Frege, or on the basis of G. Cantor's theorem. However such attempts are logically unreasonable, asthey lead to contradictions. The idea of infinity cannot be introduced a posteriori because nothing realitysays about its necessity. Hence, the axiom is just necessary. Russell treats the infinity axiom as asubstantial statement about the world.

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