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Difference between revisions of "Peano postulates"
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The arithmetic of the [[integer]]s, 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 [[Wikipedia:Giuseppe Peano|G. Peano]] in 1889; it characterises the set (class, condition) of [[natural number]]s 1, 2, 3, etc., and consists of the following '''Peano postulates''' (also called '''Peano axioms'''): | The arithmetic of the [[integer]]s, 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 [[Wikipedia:Giuseppe Peano|G. Peano]] in 1889; it characterises the set (class, condition) of [[natural number]]s 1, 2, 3, etc., and consists of the following '''Peano postulates''' (also called '''Peano axioms'''): | ||
#''1'' is a natural number | #''1'' is a natural number | ||
| − | #To each natural number | + | #To each natural number {{V|x}} there corresponds a second natural number {{V|x}}<sub>1</sub>, called the successor of {{V|x}} |
#''1'' is not the successor of any natural number | #''1'' is not the successor of any natural number | ||
| − | #From | + | #From {{V|x}}<sub>1</sub> = {{V|y}}<sub>1</sub> follows {{V|x}} = {{V|y}} |
| − | #Let | + | #Let {{V|M}} be a set of natural numbers with the following properties: |
| − | #*''1'' belongs to | + | #*''1'' belongs to {{V|M}} |
| − | #*If | + | #*If {{V|x}} belongs to {{V|M}}, then {{V|x}}<sub>1</sub> also belongs to {{V|M}}. |
| − | Then | + | Then {{V|M}} contains all natural numbers. |
In the language of these axioms, addition is defined by setting | In the language of these axioms, addition is defined by setting | ||
| − | : | + | :{{V|x}} + 1 = {{V|x}}<sub>1</sub> |
| − | : | + | :{{V|x}} + 2 = {{V|x}}<sub>1<sub>1</sub></sub> |
etc., and multiplication is defined in terms of addition: | etc., and multiplication is defined in terms of addition: | ||
| − | : | + | :{{V|ab}} = {{V|a}} + {{V|a}} + ... + {{V|a}} |
| − | where there are | + | where there are {{Vb}} 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. | 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. | ||
Revision as of 16:58, 24 October 2020
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 x1, called the successor of x
- 1 is not the successor of any natural number
- From x1 = y1 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 x1 also belongs to M.
Then M contains all natural numbers.
In the language of these axioms, addition is defined by setting
- x + 1 = x1
- x + 2 = x11
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.
