Initial Ordinal

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 omega_alphas 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 f be that isomorphism from the ordinals to the transfinite initial ordinals, then


where omega_0=omega.

See also

Ordinal Number

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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.

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