| 1. | Let denote the unary primitive recursive function given by this definition.
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| 2. | Primitive recursive functions are a defined subclass of the recursive functions.
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| 3. | With regards to ( 3 ), Kleene considers primitive recursive functions:
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| 4. | The diagonal lemma applies to theories capable of representing all primitive recursive functions.
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| 5. | A subset of these is the primitive recursive functions.
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| 6. | Such a number can therefore represent the primitive recursive function until a given n.
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| 7. | While all primitive recursive functions are provably total, the converse in not true.
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| 8. | Thus most of life goes on requiring only the " primitive recursive functions ."
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| 9. | However, all primitive recursive functions halt.
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| 10. | It allows a more natural style of expressing computation than simply using primitive recursive functions.
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