# Natural sequence

*natural number sequence*

The non-empty set in which a unary operation is defined (i.e. is a single-valued mapping of into itself) satisfying the following conditions (the Peano axioms):

1) for any ,

2) for any : If

then

3) any subset of that contains 1 and that together with any element also contains , is necessarily the whole of (axiom of induction).

The element is usually called the immediate successor of . The natural sequence is a totally ordered set. It can be proved that the conditions

where and are arbitrary elements of , define binary operations and on . The system is the system of natural numbers (cf. Natural number).

#### References

[1] | B.L. van der Waerden, "Algebra" , 1 , Springer (1967) (Translated from German) |

#### Comments

Often, the natural number sequence is started at , cf. also Natural number.

The system is the only (up to an isomorphism) system satisfying the Peano axioms.

When saying that is a totally ordered set, one refers to the total order relation defined by:

#### References

[a1] | H.C. Kennedy, "Selected works of Giuseppe Peano" , Allen & Unwin (1973) |

[a2] | E. Landau, "Grundlagen der Analysis" , Akad. Verlagsgesellschaft (1930) |

[a3] | S. MacLane, "Algebra" , Macmillan (1967) |

**How to Cite This Entry:**

Natural sequence.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Natural_sequence&oldid=16451