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According to most modern In [[mathematics]] and [[logic]] textbooks , a '''"natural " number''' is a non-negative [[integer]] such as number from the set {0, 1, 2, 3,... etc}. <ref>Unfortunately, 0 is sometimes also usually included in the list of "natural" numbers in modern textbooks (Bourbaki 1968, Halmos 1974). Older books sometimes exclude [[zero]], and as there seems to be no general agreement about whether is a long history of people thinking that zero is unnatural or not really a number. [[Bertrand Russell]] remarked on the trend to include itzero in his 1919 book. In fact, [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>Older books sometimes exclude Natural numbers were used initially for [[zerocounting]]("there are three cows in this field"), but they took on the purpose of ordering as there well ("She is a long history the 2nd fastest person alive). These are specific instances of people thinking that zero is unnatural or not really a numberthe more general notions of [[cardinality]] and [[ordinality]] which slowly become more complicated as one treats [[infinite]] numbers as well. ==Axiomatization==In the late 19th century, [[Giuseppe Peano]] (August 27, 1858 – April 20, 1932) elaborated ''the'' axiomatic system for the Natural Numbers, later named [[Peano's Axioms|Peano Axioms]] in his honor. ==References==<references/>