Chap 1 The Number System

# The following more complicated denition is therefore

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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 non-empty subset of N has a least element. Remark. The condition (WO) is called the Well-ordering 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.

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