Let  be a well ordered set. Then the set {a in A:a<k}" src="https://mathworld.wolfram.com/images/equations/InitialSegment/Inline2.gif" style="height:15px; width:80px" /> for some  is called an initial segment of  (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  and  are well ordered sets that are not order isomorphic, then exactly one of the following statements is true:

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

2.  is order isomorphic to an initial segment of 

(Dauben 1990, p. 198).

REFERENCES:

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.