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The Mathematics of Infinity: A Guide to Great Ideas, 2nd Edition by Theodore G. Faticoni

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Chapter 6

Weil-Ordered Sets

The set of real numbers images is linearly ordered. That is, given x, yimages then xy or yx. Such an ordering of a set X allows us to think of X as some part of a line. For instance, images is linearly ordered since it is a subset of images. It is the rational part of the real line. The integers, images, form a linearly ordered set, but it has the very interesting property that to each images there is a next integer or a successor to x, usually denoted by x+. One easily sees that x + 1 is the successor to x in images. The existence of a successor for each element in images is called the Well-Ordering Principle. We say that , is well-ordered. Because , is well-ordered, but is not well-ordered. In this chapter, ...

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