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Abstract : |
The object of study of the present paper may be considered as a model, in an elementary topos with a natural numbers object, of a non-classical variation of the Peano arithmetic. The new feature consists in admitting, in addition to the constant (zero) s0 2 N and the unary operation (the successor map) s1: N! N, arbitrary operations su: N u! N of arities u `between 0 and 1'. That is, u is allowed to range over subsets of a singleton set. In view of the Peano axioms, the set of natural numbers can be considered as a solution of the `equation ' X + 1 = X. Lawvere with his notion of Natural Numbers Object (NNO) gave precise meaning to this statement: here X varies, |
