Let 
and 
be well ordered sets with ordinal
numbers 
and 
.
Then 
iff 
is order isomorphic to
an initial segment of 
(Dauben 1990, p. 199). From this, it can easily be shown
that the ordinal numbers are totally
ordered by the relation. In fact, they are well
ordered by the relation.
Ordinal Comparison
See also
Well Ordered SetExplore with Wolfram|Alpha
References
Dauben, J. W. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1990.Referenced on Wolfram|Alpha
Ordinal ComparisonCite this as:
Weisstein, Eric W. "Ordinal Comparison." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OrdinalComparison.html