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