Initial Segment
المؤلف:
Dauben, J. W.
المصدر:
Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1990.
الجزء والصفحة:
...
5-1-2022
1444
Initial Segment
Let 
be a well ordered set. Then the set ![<span style=]()
{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.
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