Peano postulates

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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 x1, called the successor of x
  3. 1 is not the successor of any natural number
  4. From x1 = y1 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 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.

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