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 
.
