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)