Computability and Logic

11.2. LOGIC AND PRIMITIVE RECURSIVE FUNCTIONS 133
from earlier functions in the list by composition or primitive recursion, with the last
function f r being the function f . We introduce a language with the symbol 0, the
successor symbol ′ , and a function symbol f i of the appropriate number of places for
each of the functions f i . In the standard interpretation of the language, the domain
will be the natural numbers, 0 will denote zero, ′ will denote the successor function,
and each f i will denote f i , so that in particular f r will denote f .
The set of sentences Ɣ will consist of one or two sentence for each f i for i > 0. In
case f i is the zero function, the sentence will be
(1)
∀x f i (x) = 0.
In case f i is the successor function, the sentence will be
(2)
∀x f i (x) = x ′ .
(In this case f i will be another symbol besides ′ for the successor function; but it does
not matter if we happen to have two symbols for the same function.) In case f i is the
identity function id n k
, the sentence will be
(3)
∀x 1 ··· ∀x n f i (x 1 ,...,x n ) = x k .
If case f i is obtained from f k and f j1 , ... f jp , where j 1 , ..., j p and k are all < i,by
composition, the sentence will be
(4)
∀x f i (x) = f k (f j1 (x),...f jp (x)).
In case f i is obtained from f j and f k , where j and k are