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As the title, why do people define the segment of a well-ordering set? What's the main usage of the segment of a well-ordering set?

Well, I'm looking the book of Kunen, set theory. In chapter 1.6, he introduced the definition of pred, or we call the segment. I want to know what's his purpose of bringing this definition?

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    @Arthur: Quite by accident I just stumbled across it in Section III.5 and in Theorem IV.5.6.2012-01-12

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Ken’s immediate reason for introducing the pred operator at that point is to be able to state Theorem I.6.3, which is fundamental to any understanding of well-orders: if $\langle X,\le\rangle$ and $\langle Y,\preceq\rangle$ are well-orders, then either they are isomorphic, or one is isomorphic to a proper initial segment of the other. In the very next section he goes on to construct the ordinals $-$ transitive sets $\alpha$ such that $\langle \alpha,\in\rangle$ is a well-order $-$ and shows that every well-order is isomorphic to one of them. This improves on Theorem I.6.3, because if $\alpha$ and $\beta$ are ordinals, then either they are equal (not just isomorphic), or one of them is a proper initial segment of the other (not just isomorphic to one). Moreover, each ordinal is a (necessarily proper!) initial segment of the proper class ON of ordinals. Thus, the notion of initial segment is intimately bound up with the concept of a well-order, which in turn is fundamental to set theory.

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    It's very enlightening for a beginner of learning set theory!2012-01-12