History of Natural number

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 Revision 24 . . (edit) December 13, 2001 8:59 am by Bryan Derksen [typo] Revision 23 . . (edit) December 13, 2001 8:59 am by Bryan Derksen [unicode warning...] Revision 22 . . December 9, 2001 11:18 am by AxelBoldt [reverting to non-unicode] Revision 21 . . (edit) December 9, 2001 9:13 am by Bryan Derksen [unicode symbol] Revision 20 . . November 17, 2001 11:38 am by AxelBoldt [Every monoid is associative. Reverting.] Revision 19 . . November 17, 2001 11:09 am by (logged).163.213.xxx Revision 18 . . November 13, 2001 7:31 am by AxelBoldt [streamline paragraph about zero's status]

Difference (from prior major revision) (minor diff, author diff)

Changed: 1c1
 A natural number is any of the numbers 0, 1, 2, 3... that can be used to measure the size of finite sets.
 A natural number is any of the numbers 0, 1, 2, 3... that can be used to measure the size of finite sets.

Changed: 9c9
 The Peano postulates essentially uniquely describe the set ℕ of natural numbers:
 The Peano postulates essentially uniquely describe the set of natural numbers, which is denoted by N (or more properly by the unicode character ℕ if your browser supports unicode display).

Changed: 21c21
 One can inductively define an addition on the natural numbers by requiring a + (b + 1) = (a + b) + 1. This turns the natural numbers (ℕ,+) into a commutative monoid with neutral element 0, the so-called free monoid with one generator. This monoid satisfies the cancellation property and can therefore be embedded in a group. The smallest group containing the natural numbers is the integers.
 One can inductively define an addition on the natural numbers by requiring a + (b + 1) = (a + b) + 1. This turns the natural numbers (N, +) into a commutative monoid with neutral element 0, the so-called free monoid with one generator. This monoid satisfies the cancellation property and can therefore be embedded in a group. The smallest group containing the natural numbers is the integers.

Changed: 23c23
 Analogously, a multiplication * can be defined via a * (b + 1) = ab + a. This turns (ℕ,*) into a commutative monoid; addition and multiplication are compatible which is expressed in the [distribution law]?:
 Analogously, a multiplication * can be defined via a * (b + 1) = ab + a. This turns (N, *) into a commutative monoid; addition and multiplication are compatible which is expressed in the [distribution law]?:

Changed: 34c34
 Natural numbers can be used for two purposes: to describe the position of an element in an ordered sequence, which is generalized by the concept of ordinal number, and to specify the size of a finite set, which is generalized by the concept of cardinal number. In the finite world, these two concepts coincide: the finite ordinals are equal to ℕ as are the finite cardinals. When moving beyond the finite, however, the two concepts diverge.
 Natural numbers can be used for two purposes: to describe the position of an element in an ordered sequence, which is generalized by the concept of ordinal number, and to specify the size of a finite set, which is generalized by the concept of cardinal number. In the finite world, these two concepts coincide: the finite ordinals are equal to N as are the finite cardinals. When moving beyond the finite, however, the two concepts diverge.

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