Successor function

For other uses, see Successor.

In mathematics, the successor function or successor operation is a primitive recursive function defined by

$S(n)=n+1.\,$

We have S(1) = 1 + 1 = 2, S(S(1)) = 2 + 1 = 3, ...

The successor function is used in the Peano axioms which define the natural numbers. As such, it is not defined by addition, but rather is used to define all natural numbers beyond 0, as well as addition. For example, 1 is defined to be S(0), and addition on natural numbers is defined recursively by:

$m+0=m$

$m+S(n)=S(m)+n$

This yields 5 + 2 = 5 + S(1) = S(5) + 1 = 6 + 1 = 6 + S(0) = S(6) + 0 = 7 + 0 = 7

It is the level-0 foundation of the infinite hierarchy of hyperoperations (used to build addition, multiplication, exponentiation, etc.).

It is also one of the primitive functions used in the characterization of computability by recursive functions.

See also

• successor ordinal
• successor cardinal