# 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 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).

How to Cite This Entry:
Natural sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Natural_sequence&oldid=16451
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