Successor function

The successor function is the function \(f(x) = x + 1\).authority.20110803100540531 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 sometimes called a successor ordinal if it is in the form \(\alpha = \beta+1\).
== Turing Machine code ==
0 1 * r 0
0 _ 1 * halt
Explanation: In a Turing machine, the integer \(n\) is represented by a string of \(n\) consecutive 1s. The above program then takes a string of \(n\) consecutive 1s and replaces it with \(n+1\) consecutive 1s, by searching for the leftmost empty cell and then placing a 1 in it.
== Related functions ==
In some fields, the terms "zeration" and "hyper0" are ambiguous and can also refer to a family of binary operatorsZerationwhat-operation-repeated-n-times-results-in-the-addition-operator which satisify \(f(f(f(a,a),a),a\text{...})\;(\textrm{with}\; b\; ``a\!"\textrm s) = a+b\), including the function:
\(f(a,b) = a + 1,\ a\ >\ b \\
f(a,b) = b + 1 ,\ a\ <\ b \\ 
f(a,b) = a + 2 = b + 2 ,\ a\ =\ b \\
\)
== Source ==
== See also ==
ja:後者関数
zh:后继运算
Category:Functions
Category:Unary operators