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Unformatted text preview: numbers, usually given by N = {1, 2, 3, . . .}. However, this deﬁnition does not bring out some of the main properties of the set N in a natural way. The following more complicated deﬁnition is therefore sometimes preferred. Definition. The set N of all natural numbers is deﬁned by the following four conditions: (N1) 1 ∈ N. (N2) If n ∈ N, then the number n + 1, called the successor of n, also belongs to N. (N3) Every n ∈ N other than 1 is the successor of some number in N. (WO) Every nonempty subset of N has a least element. Remark. The condition (WO) is called the Wellordering principle. To explain the signiﬁcance of each of these four requirements, note that the conditions (N1) and (N2) together imply that N contains 1, 2, 3, . . . . However, these two conditions alone are insuﬃcient to exclude from N numbers such as 5.5. Now, if N contained 5.5, then by condition (N3), N must also contain 4.5, 3.5, 2.5, 1.5, 0.5, −0.5, −1.5, −2.5, . . . , and so would not have a least element. We therefore ex...
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This note was uploaded on 02/01/2009 for the course MATH 2343124 taught by Professor Staff during the Fall '08 term at UCSD.
 Fall '08
 staff
 Math, Calculus

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