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In [[mathematics]], a '''"natural" number''' is a number from the set {0,1,2,...}.<ref>0 is usually included in the list of natural numbers in modern textbooks (Bourbaki 1968, Halmos 1974). Older books sometimes exclude [[zero]], as there is a long history of people thinking that zero is unnatural or not really a number. [[Bertrand Russell]] remarked on the trend to include zero in his 1919 book.[http://jeff560.tripod.com/n.html] Ribenboim (1996) states "Let P be a set of natural numbers; whenever convenient, it may be assumed that 0 in P." [http://mathworld.wolfram.com/NaturalNumber.html (Wolfram)] </ref> Natural numbers were used initially for [[counting]] ("there are three cows in this field"), but they took on the purpose of ordering as well ("She is the 2nd fastest person alive). These are specific instances of the more general notions of [[cardinality]] and [[ordinality]] which slowly become more complicated as one treats [[infinite]] numbers as well. The set of natural numbers is [[countable]]- via [[bijection]], this property can be used to prove the countability of the [[integer]]s and [[rational number]]s.

In [[mathematics]], a '''"natural" number''' is a number from the set {1,2,3,...}.<ref>0 is usually included in the list of natural numbers in modern textbooks (Bourbaki 1968, Halmos 1974). Older books sometimes exclude [[zero]], as there is a long history of people thinking that zero is unnatural or not really a number. [[Bertrand Russell]] remarked on the trend to include zero in his 1919 book.[http://jeff560.tripod.com/n.html] Ribenboim (1996) states "Let P be a set of natural numbers; whenever convenient, it may be assumed that 0 in P." [http://mathworld.wolfram.com/NaturalNumber.html (Wolfram)] </ref> Natural numbers were used initially for [[counting]] ("there are three cows in this field"), but they took on the purpose of ordering as well ("She is the 2nd fastest person alive). These are specific instances of the more general notions of [[cardinality]] and [[ordinality]] which slowly become more complicated as one treats [[infinite]] numbers as well. The set of natural numbers is [[countable]]- via [[bijection]], this property can be used to prove the countability of the [[integer]]s and [[rational number]]s.

==Axiomatization==

==Axiomatization==