An ordinal number is called an initial ordinal if every smaller ordinal has a smaller cardinal number. The ω_α s ordinal numbers are just the transfinite initial ordinals. 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 ω_α = f(α), where ω_0 = ω.

ordinal number