Initial Segment

Let (A,<=) be a well ordered set. Then the set {a in A:a<k} for some k in A is called an initial segment of A (Rubin 1967, p. 161; Dauben 1990, pp. 196-197; Moore 1982, pp. 90-91). This term was first used by Cantor, who also proved that if (A,<=) and (B,<=) are well ordered sets that are not order isomorphic, then exactly one of the following statements is true:

1. A is order isomorphic to an initial segment of B, or

2. B is order isomorphic to an initial segment of A

(Dauben 1990, p. 198).

See also

Well Ordered Set

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Dauben, J. W. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1990.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.

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Initial Segment

Cite this as:

Weisstein, Eric W. "Initial Segment." From MathWorld--A Wolfram Web Resource.

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