SucC

Succ, short for successor, is a function in mathematics and computer science that assigns to each natural number its immediate successor. Commonly written as succ(n) or S(n), it yields n+1. In the Peano axioms for the natural numbers, succ is a primitive operation alongside the constant 0. The axioms state that 0 is a natural number, every natural number has a unique successor, 0 is not the successor of any number, and distinct numbers have distinct successors. From succ, the usual arithmetic operations and the order structure of the natural numbers are defined by recursion and induction.

Properties of succ include that it is injective: if succ(a) = succ(b) then y a = b. In the standard model of natural numbers, no number is the successor of 0, and every number except 0 has a predecessor, which is the inverse of succ where defined. In ordinal arithmetic, the successor of an ordinal α is α+1; ordinals that are not successor ordinals are called limit ordinals.

In programming and formal languages, succ is used as a function that returns the next value in an ordered type. In Haskell, for example, succ x yields the successor of x within the Enum class, with an error if x is the maximum bound. In other languages, succ is often realized as the operation of adding 1 to an integer or advancing to the next element in an enumeration.

Overall, succ is a foundational concept for constructing natural numbers, defining arithmetic, and enabling iteration in algorithms.