Abstract
Abstract The axiom system Adoes not provide an adequate foundation for the whole of the arithmetic of real numbers, because-as we saw in Section 60-numerous theorems of this discipline cannot be deduced from the axioms of this system, and also for another reason, which is analogous and not less important: a number of concepts belonging to the field of arithmetic are not definable if we utilize only the primitive terms which occur in System A.For example, on its basis we shall never be able to define the symbols of multiplication or division, or such symbols as “1”, “2”, and so on. The following question then arises in a natural way: how can we transform or supplement our system of axioms and primitive terms in order to arrive at a satisfactory basis for constructing the full arithmetic of real numbers? This problem can be solved in a variety of ways. Two quite different methods of solution will be sketched here.
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