| 11. | This is a primitive recursive function.
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| 12. | Every primitive recursive function is total recursive, but not all total recursive functions are primitive recursive.
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| 13. | The functions that can be programmed in the LOOP programming language are exactly the primitive recursive functions.
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| 14. | In the late 19th century, Leopold Kronecker formulated notions of computability, defining primitive recursive functions.
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| 15. | Defining primitive recursive functions in this manner is not possible in PRA, because it lacks quantifiers.
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| 16. | Primitive recursive functions tend to correspond very closely with our intuition of what a computable function must be.
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| 17. | Truncated subtraction is useful in contexts such as primitive recursive functions, which are not defined over negative numbers.
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| 18. | This means that the-th definition of a primitive recursive function in this enumeration can be effectively determined from.
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| 19. | All other primitive recursive predicates can be defined using these two primitive recursive functions and quantification over all natural numbers.
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| 20. | A set of axioms is primitive recursive if there is a primitive recursive function that decides membership in the set.
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