Importantly, all the formulas in the proof can be defined by primitive recursive functions, which themselves first-order Peano arithmetic.
32.
The "'concepts "'are extremely important, but are already discussed at primitive recursive function and computable function.
33.
And any function that can be computed by a Turing machine in a running time bounded by a primitive recursive function is itself primitive recursive.
34.
Some examples include the first primitive recursive function that results in complexity, the smallest universal Turing Machine, and the shortest axiom for propositional calculus.
35.
The primitive recursive functions are closely related to mathematical finitism, and are used in several contexts in mathematical logic where a particularly constructive system is desired.
36.
The definitions of elementary recursive functions are the same as for primitive recursive functions, except that primitive recursion is replaced by bounded summation and bounded product.
37.
In particular it is not primitive recursive, but it is also far larger than standard examples of non primitive recursive functions such as the Ackermann function.
38.
The language of PRA can express arithmetic propositions involving natural numbers and any primitive recursive function, including the operations of addition, multiplication, and exponentiation.
39.
:" A recursive function ( relation ) in the sense of G�del . . . will now be called a primitive recursive function ( relation ).
40.
While all primitive recursive functions are total, this is not true of partial recursive functions; for example, the minimisation of the successor function is undefined.
