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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 ''x'' there corresponds a second natural number ''x<sub>1</sub>'', called the successor of ''x''
+
#To each natural number {{V|x}} there corresponds a second natural number {{V|x}}<sup>+</sup>, 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 ''x<sub>1</sub> = y<sub>1</sub>'' follows ''x = y''
+
#From {{V|x}}<sup>+</sup> = {{V|y}}<sup>+</sup> follows {{V|x}} = {{V|y}}
#Let ''M'' be a set of natural numbers with the following properties:
+
#Let {{V|M}} be a set of natural numbers with the following properties:
#*''1'' belongs to ''M''
+
#*''1'' belongs to {{V|M}}
#*If ''x'' belongs to ''M'', then ''x<sub>1</sub>'' also belongs to ''M''.
+
#*If {{V|x}} belongs to {{V|M}}, then {{V|x}}<sup>+</sup> also belongs to {{V|M}}.
  
Then ''M'' contains all natural numbers.
+
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
:''x'' + 1 = ''x<sub>1</sub>''
+
:{{V|x}} + 1 = {{V|x}}<sup>+</sup>
:''x'' + 2 = ''x<sub>1<sub>1</sub></sub>
+
:{{V|x}} + 2 = {{V|x}}<sup>+<sup>+</sup></sup>
 
etc., and multiplication is defined in terms of addition:
 
etc., and multiplication is defined in terms of addition:
:''ab'' = ''a'' + ''a'' + ... + ''a''
+
:{{V|ab}} = {{V|a}} + {{V|a}} + ... + {{V|a}}
where there are ''b'' terms on the right hand side.
+
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  ''&lt;'' 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  ''&lt;'' can be introduced as required. Finally, zero and the negative integers can be defined in terms of the natural numbers.

Latest revision as of 17:04, 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. 1 is a natural number
  2. To each natural number x there corresponds a second natural number x+, called the successor of x
  3. 1 is not the successor of any natural number
  4. From x+ = y+ follows x = y
  5. 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.

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