Abstract
A partially ordered set may not have a smallest element, and, even if it has one, it is perfectly possible that some subset will fail to have one. A partially ordered set is called well ordered (and its ordering is called a well ordering) if every non-empty subset of it has a smallest element. One consequence of this definition, worth noting even before we look at any examples and counterexamples, is that every well ordered set is totally ordered. Indeed, if x and y are elements of a well ordered set, then {x, y} is a non-empty subset of that well ordered set and has therefore a first element; according as that first element is x or y, we have x ≦ y or y ≦ x.
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© 1974 Springer Science+Business Media New York
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Halmos, P.R. (1974). Well Ordering. In: Naive Set Theory. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-1645-0_17
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DOI: https://doi.org/10.1007/978-1-4757-1645-0_17
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-90104-6
Online ISBN: 978-1-4757-1645-0
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