 Successor function

The successor function on the natural numbers is the function $S(n)=n+1$. With $0$ as the initial natural number, its repeated application produces $0,S(0),S(S(0)),\dots$.

In recursion theory, the successor function is an initial function of recursion theory. Together with the other initial functions and the permitted operations, it helps generate primitive recursive functions. For example, addition can be specified recursively by $a+0=a$ and $a+S(b)=S(a+b)$. This description builds addition from a base case and a successor step, matching the structure used in mathematical induction.

In mathematics, particularly in the context of set theory and number theory, the successor function is used to define the concept of "next" numbers in a sequence. For natural numbers, the successor function takes a natural number \( n \) and gives the next natural number \( n + 1 \). For example: - If \( n = 0 \), then the successor of \( n \) (often denoted as \( S(n) \)) is 1.