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)

Comments

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)