such that if is true then returns the output of the function with index on input .

Because Kleene's formalism attaches a number of inputs to each function, the predicate can only be used for functions that take one input. There are additional predicates for functions with multiple inputs; the relation

is true if encodes a halting computation of the function with index on the inputs .

Like , all functions are primitive recursive.Because of this, any theory of arithmetic that is able to represent every

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