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 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 .

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.