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The aim of this chapter is to show how to develop an arithmetic of natural numbers within the pure theory of combinators. In section 2 we shall give a simple proof of the combinatory definability of all partial recursive functions. This result is essentially due to Kleene 1936, who developed arithmetic within the λ—I-calculus without an analog to the combinator K. In the presence of K it is possible to simplify the proof a bit. The first to develop arithmetic within the theory of combinators was Curry 1941.
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