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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]
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.
Because its elements are related to the previous elements in a straightforward way, they are often defined using recursion. A drawing of the first 75 terms of Recamán's sequence, according with the method of visualization shown in the Numberphile video The Slightly Spooky Recamán Sequence [3]
Recursive drawing of a Sierpiński Triangle through turtle graphics. In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. [1] [2] Recursion solves such recursive problems by using functions that call themselves from within their own code ...
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 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.
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.
The Cantor pairing function assigns one natural number to each pair of natural numbers Graph of the Cantor pairing function The Cantor pairing function is a primitive recursive pairing function π : N × N → N {\displaystyle \pi :\mathbb {N} \times \mathbb {N} \to \mathbb {N} }