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Primitive recursive arithmetic (PRA) is a quantifier-free formalization of the natural numbers. It was first proposed by Norwegian mathematician Skolem (1923) , [ 1 ] as a formalization of his finitistic conception of the foundations of arithmetic , and it is widely agreed that all reasoning of PRA is finitistic.
An example of a primitive recursive programming language is one that contains basic arithmetic operators (e.g. + and −, or ADD and SUBTRACT), conditionals and comparison (IF-THEN, EQUALS, LESS-THAN), and bounded loops, such as the basic for loop, where there is a known or calculable upper bound to all loops (FOR i FROM 1 TO n, with neither i ...
Gentzen's theory obtained by adding quantifier-free transfinite induction to primitive recursive arithmetic proves the consistency of first-order Peano arithmetic (PA) but does not contain PA. For example, it does not prove ordinary mathematical induction for all formulae, whereas PA does (since all instances of induction are axioms of PA).
A recursive step — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ancestor. One's ancestor is either: One's parent (base case), or; One's parent's ancestor (recursive step). The Fibonacci sequence is another classic example of recursion: Fib(0) = 0 as ...
Elementary recursive arithmetic (ERA) is a subsystem of primitive recursive arithmetic (PRA) in which recursion is restricted to bounded sums and products. This also has the same Π 2 0 {\displaystyle \Pi _{2}^{0}} sentences as EFA, in the sense that whenever EFA proves ∀x∃y P ( x , y ), with P quantifier-free, ERA proves the open formula P ...
The primitive recursive functions are a subset of the total recursive functions, which are a subset of the partial recursive functions. For example, the Ackermann function can be proven to be total recursive, and to be non-primitive. Primitive or "basic" functions: Constant functions C k n: For each natural number n and every k
For example, in primitive recursive arithmetic any computable function that is provably total is actually primitive recursive, while Peano arithmetic proves that functions like the Ackermann function, which are not primitive recursive, are total.
For example, (), is a well-formed formula of second-order arithmetic that is arithmetical, has one free set variable X and one bound individual variable n (but no bound set variables, as is required of an arithmetical formula)—whereas (<) is a well-formed formula that is not arithmetical, having one bound set variable X and one bound ...