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This may be a slightly vague question but if one defines a function (of some arity) recursively on the natural numbers, the "simplest" examples are things like addition, multiplication, or factorial.

How do these functions fit into a general sense of defining recursively functions on the natural numbers? Starting only from the successor operator.

What indeed exactly is an "arithmetic function"?

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    Perhaps you're thinking of primitive recursive functions.2012-12-13
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    Yes, perhaps now a button to "restate" the question might be nice, given that I know or recall a bit more now. Obviously if I edit it loads, it will make the ensuing answers that were helpful, look silly.2012-12-13

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