# Natural sequence

natural number sequence

The non-empty set $\mathbf N = \{ 1 , 2 , . . . \}$ in which a unary operation $S$ is defined (i.e. $S$ is a single-valued mapping of $\mathbf N$ into itself) satisfying the following conditions (the Peano axioms):

1) for any $a \in \mathbf N$,

$$1 \neq Sa;$$

2) for any $a, b \in \mathbf N$: If

$$Sa = Sb,$$

then

$$a = b;$$

3) any subset of $\mathbf N$ that contains 1 and that together with any element $a$ also contains $Sa$, is necessarily the whole of $\mathbf N$( axiom of induction).

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

$$a + 1 = Sa,\ \ a + Sb = S ( a + b),$$

$$a \cdot 1 = a,\ a \cdot Sb = ab + a,$$

where $a$ and $b$ are arbitrary elements of $\mathbf N$, define binary operations $(+)$ and $( \cdot )$ on $\mathbf N$. The system $\langle \mathbf N, +, \cdot , 1 \rangle$ is the system of natural numbers (cf. Natural number).

#### References

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

Often, the natural number sequence is started at $0$, cf. also Natural number.

The system $( \mathbf N , S )$ is the only (up to an isomorphism) system satisfying the Peano axioms.

When saying that $( \mathbf N , S )$ is a totally ordered set, one refers to the total order relation $<$ defined by:

$$\neg ( a < 1 ) ,$$

$$a < S b \iff a < b \textrm{ or } a = b .$$

#### 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=47949
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