Successor function

The successor function is the function \(f(x) = \textrm{Least natural number greater than }x\), often characterized by \(f(x)=x+1\). All early formal attempts at defining the largest number were based around finding different methods for using recursion on the successor function. The successor function is still important for defining large numbers as many functions like busy beaver and Rayo use non-intuitive definitions of natural numbers, for which the successor function must be defined in order to establish what they are calculating is actually a number. The successor function is equal to \(f_0\) of the fast-growing hierarchy.

Nomenclature
It is usually considered the 0th hyperoperation, and is thus sometimes called zeration or incrementation, as well as hyper0.

Successor function is commonly denoted by \(S(n)\) (not to be confused with maximum shifts function.)

An ordinal \(\alpha\) is called a successor ordinal if it is of the form \(\alpha = \beta+1\).

Related functions
In some fields, the terms "zeration" and "hyper0" are ambiguous and can also refer to a family of binary operators which satisify \(f(f(f(a,a),a),a\text{...})\;(\textrm{with}\; b\; ``a\!"\textrm s) = a+b\), including the function: \(f(a,b) = a + 1,\ a\ >\ b \\ f(a,b) = b + 1 ,\ a\ <\ b \\ f(a,b) = a + 2 = b + 2 ,\ a\ =\ b \\ \)