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# Peano postulates

The arithmetic of the integers, like the geometry of the plane, can be made to depend on a few axioms, in the sense that everything else follows from them by accepted logical rules. One such set of axioms was given by G. Peano in 1889; it characterises the set (class, condition) of natural numbers 1, 2, 3, etc., and consists of the following **Peano postulates** (also called **Peano axioms**):

*1*is a natural number- To each natural number
`x`there corresponds a second natural number`x`^{+}, called the successor of`x` *1*is not the successor of any natural number- From
`x`^{+}=`y`^{+}follows`x`=`y` - Let
`M`be a set of natural numbers with the following properties:*1*belongs to`M`- If
`x`belongs to`M`, then`x`^{+}also belongs to`M`.

Then `M` contains all natural numbers.

In the language of these axioms, addition is defined by setting

`x`+ 1 =`x`^{+}`x`+ 2 =`x`^{++}

etc., and multiplication is defined in terms of addition:

`ab`=`a`+`a`+ ... +`a`

where there are `b` terms on the right hand side.

The usual rules of algebra can then be deduced, as they apply to the natural numbers, and the inequality symbol *<* can be introduced as required. Finally, zero and the negative integers can be defined in terms of the natural numbers.