Inmathematics, thesuccessor function orsuccessor operation sends anatural number to the next one. The successor function is denoted by${\displaystyle S}$, so${\displaystyle S(n)=n+1}$. For example,${\displaystyle S(1)=2}$ and${\displaystyle S(2)=3}$. The successor function is one of the basic components used to build aprimitive recursive function.

Successor operations are also known aszeration in the context of a zerothhyperoperation. In this context, the extension of zeration isaddition, which is defined as repeated succession.

The successor function is part of theformal language used to state thePeano axioms, which formalise the structure of the natural numbers. In this formalisation, the successor function is a primitive operation on the natural numbers, in terms of which the standard natural numbers and addition are defined.^[1] 1 is defined to be${\displaystyle S(0)}$, 2 is${\displaystyle S(1)}$, etc.; and addition on natural numbers is defined recursively by:

${\displaystyle m+0=m}$

${\displaystyle m+S(n)=S(m+n)}$

This can be used to compute the addition of any two natural numbers. For example:

${\displaystyle 5+2}$

${\displaystyle =5+S(1)}$

${\displaystyle =5+S(S(0))}$

${\displaystyle =S(5+S(0))}$

${\displaystyle =S(S(5+0))}$

${\displaystyle =S(S(5))}$

${\displaystyle =S(6)}$

${\displaystyle =7}$.

Severalconstructions of the natural numbers within set theory have been proposed. For example,John von Neumann constructs the number 0 as theempty set and the successor of${\displaystyle n}$ as the set${\displaystyle n\cup \{n\}}$. Theaxiom of infinity then guarantees the existence of a set that contains 0 and isclosed with respect to${\displaystyle S}$. The smallest such set is denoted by${\displaystyle \mathbb{N}}$, and its members are callednatural numbers.^[2]

The successor function is the level-0 foundation of the infiniteGrzegorczyk hierarchy ofhyperoperations, used to buildaddition,multiplication,exponentiation,tetration, etc. It was studied in 1986 in an investigation involving generalization of the pattern for hyperoperations.^[3]

It is also one of the primitive functions used in the characterization ofcomputability byrecursive functions.

• Successor ordinal
• Successor cardinal
• Increment and decrement operators
• Sequence

• Paul R. Halmos (1968).Naive Set Theory. Nostrand.

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