An ordinal number is called an initial ordinal if every smaller ordinal has a smaller cardinal number
(Moore 1982, p. 248; Rubin 1967, p. 271). The s ordinal numbers are just the transfinite initial
ordinals (Rubin 1967, p. 272).

This proper class can be well ordered and put into one-to-one correspondence with the ordinal numbers . For any two well
ordered sets that are order isomorphic , there
is only one order isomorphism between them. Let be that isomorphism from the ordinals to the transfinite initial
ordinals, then

where .

See also Ordinal Number
Explore with Wolfram|Alpha
References 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. Referenced
on Wolfram|Alpha Initial Ordinal
Cite this as:
Weisstein, Eric W. "Initial Ordinal."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/InitialOrdinal.html

Subject classifications