User blog comment:Edwin Shade/Enumerating the Countable Ordinals/@comment-25337554-20171208234807/@comment-25337554-20171209002843

Thank you for the response. I also find another fact:

If you have an enumeration up to an ordinal α, you can make a fundamental sequence from that:

List all "valid" numbers and corresponding ordinals. Let the ordinal sequence {a_n}. Then:

α[0]=a_0

if α[n]=a_t, then α[n+1]=a_u s.t. t>u∧(∀x<u, a_x<a_u)