Natural sequence

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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


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).


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


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:


[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:
This article was adapted from an original article by A.A. BukhshtabV.I. Nechaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article