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 ,
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2) for any : If
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then
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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
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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:
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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) |
Natural sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Natural_sequence&oldid=39750