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Well Ordered Set

A totally ordered set (A,<=) is said to be well ordered (or have a well-founded order) iff every nonempty subset of A has a least element (Ciesielski 1997, p. 38; Moore 1982, p. 2; Rubin 1967, p. 159; Suppes 1972, p. 75). Every finite totally ordered set is well ordered. The set of integers Z, which has no least element, is an example of a set that is not well ordered.

An ordinal number is the order type of a well ordered set.

See also

Axiom of Choice, Hilbert's Problems, Initial Segment, Monomial Order, Ordinal Number, Order Type, Subset, Well Ordering Principle

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References

Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997.Ferreirós, J. "Well-Ordered Sets." §8.4 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkhäuser, pp. 274-278, 1999.Moore, G. H. Zermelo's Axiom of Choice: Its Origin, Development, and Influence. New York: Springer-Verlag, 1982.Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967.Séroul, R. Programming for Mathematicians. Berlin: Springer-Verlag, pp. 22-23, 2000.Suppes, P. Axiomatic Set Theory. New York: Dover, 1972.

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Well Ordered Set

Cite this as:

Weisstein, Eric W. "Well Ordered Set." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WellOrderedSet.html

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