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
| omega_alpha=f(alpha), |
where omega_0=omega.
See also
Ordinal NumberExplore with Wolfram|Alpha
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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 OrdinalCite this as:
Weisstein, Eric W. "Initial Ordinal." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/InitialOrdinal.html