The key property of the LOOP language is that the functions it can compute are exactly the primitive recursive functions.
22.
There is a characterization of the primitive recursive functions as a subset of the total recursive functions using the Ackermann function.
23.
:The primitive recursive functions of one argument ( i . e ., unary functions ) can be computably enumerated.
24.
In first-order arithmetic, the only primitive recursive functions that need to be explicitly axiomatized are addition and multiplication.
25.
Is " more basic ", because it closely resembles the Peano axioms and the operators of the primitive recursive functions.
26.
The primitive recursive functions are a subset of the total recursive functions, which are a subset of the partial recursive functions.
27.
The demonstration will use a " successor " counter machine model closely related to the Peano Axioms and the primitive recursive functions.
28.
Finally, there are symbols for any primitive recursive functions with corresponding defining equations, as in Skolem's system above.
29.
By using G�del numberings, the primitive recursive functions can be extended to operate on other objects such as integers and rational numbers.
30.
It is possible to define the arithmetical hierarchy of formulas using a language extended with a function symbol for each primitive recursive function.
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