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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.
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 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.
In Zermelo–Fraenkel (ZF) set theory, the natural numbers are defined recursively by letting 0 = {} be the empty set and n + 1 (the successor function) = n ∪ {n} for each n. In this way n = {0, 1, …, n − 1} for each natural number n. This definition has the property that n is a set with n elements.
The natural numbers 𝐍 are an NNO where z is a function from a singleton to 𝐍 whose image is zero, and s is the successor function. (We could actually allow z to pick out any element of 𝐍, and the resulting NNO would be isomorphic to this one.) One can prove that the diagram in the definition commutes using mathematical induction.
In mathematical logic, Gödel's β function is a function used to permit quantification over finite sequences of natural numbers in formal theories of arithmetic. The β function is used, in particular, in showing that the class of arithmetically definable functions is closed under primitive recursion, and therefore includes all primitive recursive functions.
The primitive recursive functionals are the smallest collection of objects of finite type such that: The constant function f(n) = 0 is a primitive recursive functional; The successor function g(n) = n + 1 is a primitive recursive functional; For any type σ×τ, the functional K(x σ, y τ) = x is a primitive recursive functional