mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the successor function or successor operation sends a

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor function is one of the basic components used to build a primitive recursive function. Successor operations are also known as zeration in the context of a zeroth hyperoperation: H₀(''a'', ''b'') = 1 + ''b''. In this context, the extension of zeration is

addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol, +) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication, and Division (mathematics), divis ...

, which is defined as repeated succession.

Overview

The successor function is part of the formal language used to state the Peano 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. For example, 1 is defined to be ''S''(0), and addition on natural numbers is defined recursively by: : This can be used to compute the addition of any two natural numbers. For example, 5 + 2 = 5 + ''S''(1) = ''S''(5 + 1) = ''S''(5 + ''S''(0)) = ''S''(''S''(5 + 0)) = ''S''(''S''(5)) = ''S''(6) = 7. Several constructions of the natural numbers within set theory have been proposed. For example,

John von Neumann John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...

constructs the number 0 as the

empty set In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...

, and the successor of ''n'', ''S''(''n''), as the set ''n'' ∪ . The axiom of infinity then guarantees the existence of a set that contains 0 and is closed with respect to ''S''. The smallest such set is denoted by N, and its members are called natural numbers.Halmos, Chapter 11 The successor function is the level-0 foundation of the infinite Grzegorczyk hierarchy of hyperoperations, used to build

multiplication Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...

exponentiation In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...

, tetration, etc. It was studied in 1986 in an investigation involving generalization of the pattern for hyperoperations. It is also one of the primitive functions used in the characterization of computability by recursive functions.

References

* Mathematical logic Arithmetic Logic in computer science {{mathlogic-stub

Overview

See also

References