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The successor function is part of the formal language used to state the Peano axioms, which formalise the structure of the natural numbers.In this formalisation, the successor function is a primitive operation on the natural numbers, in terms of which the standard natural numbers and addition are defined. [1]
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. Primitive recursive arithmetic (PRA), a formal axiom system for the natural numbers and the primitive recursive functions on them, is often used for this purpose.
Structural recursion includes nearly all tree traversals, including XML processing, binary tree creation and search, etc. By considering the algebraic structure of the natural numbers (that is, a natural number is either zero or the successor of a natural number), functions such as factorial may also be regarded as structural recursion.
In mathematics and computer science, Recamán's sequence [1] [2] is a well known sequence defined by a recurrence relation.Because its elements are related to the previous elements in a straightforward way, they are often defined using recursion.
Many mathematical axioms are based upon recursive rules. For example, the formal definition of the natural numbers by the Peano axioms can be described as: "Zero is a natural number, and each natural number has a successor, which is also a natural number." [2] By this base case and recursive rule, one can generate the set of all natural numbers.
The entire set of natural numbers is computable. Each natural number (as defined in standard set theory) is computable; that is, the set of natural numbers less than a given natural number is computable. The subset of prime numbers is computable. A recursive language is a computable subset of a formal language.
For example, the natural numbers (see Peano arithmetic) may be defined by the Haskell datatype: data Nat = Zero | Succ Nat In type theory, we would say: n a t = μ α .1 + α {\displaystyle nat=\mu \alpha .1+\alpha } where the two arms of the sum type represent the Zero and Succ data constructors.
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.