and recursive defining equations for every primitive recursive function as desired. For instance, the most common characterization of the primitive recursive functions is as the 0 constant and successor function closed under projection, composition and primitive recursion. So for a (n+1)-place function f defined by primitive recursion over a n-place base function g and (n+2)-place iteration function h there would be the defining equations:
It is possible to formalise PRA in such a way that it has no logical connectives at all—a sentence of PRA is just an equation between two terms. In this setting a term is a primitive recursive function of zero or more variables. Curry (1941) gave the first such system. The rule of induction in Curry's system was unusual. A later refinement was given by Goodstein (1954). The rule of induction in Goodstein's system is:
Here x is a variable, S is the successor operation, and F, G, and H are any primitive recursive functions which may have parameters other than the ones shown. The only other inference rules of Goodstein's system are substitution rules, as follows:
Here A, B, and C are any terms (primitive recursive functions of zero or more variables). Finally, there are symbols for any primitive recursive functions with corresponding defining equations, as in Skolem's system above.
In this way the propositional calculus can be discarded entirely. Logical operators can be expressed entirely arithmetically, for instance, the absolute value of the difference of two numbers can be defined by primitive recursion:
Thus, the equations x=y and are equivalent. Therefore the equations and express the logical conjunction and disjunction, respectively, of the equations x=y and u=v. Negation can be expressed as .
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