>>916

　>>808(参考)(再掲)>>631より
en.wikipedia.org/wiki/Well-ordering_theorem
Well-ordering theorem
Proof from axiom of choice
The well-ordering theorem follows from the axiom of choice as follows.[9]
Let the set we are trying to well-order be A, and let f be a choice function for the family of non-empty subsets of A.　
For every ordinal α, define an element aα that is in A by setting
aα= f(A∖{aξ∣ξ<α})
if this complement A∖{aξ∣ξ<α} is nonempty, or leave aα undefined if it is.
That is, aα is chosen from the set of elements of A that have not yet been assigned a place in the ordering (or undefined if the entirety of A has been successfully enumerated).
Then the order < on A defined by aα<aβ if and only if α<β (in the usual well-order of the ordinals) is a well-order of A as desired, of order type sup{α∣aα is defined}.
Notes
9＾ Jech, Thomas (2002). Set Theory (Third Millennium Edition). Springer. p. 48. ISBN 978-3-540-44085-7.
(引用終り)

ここで
”Then the order < on A defined by aα<aβ if and only if α<β (in the usual well-order of the ordinals) is a well-order of A as desired,”
の部分、”the order < on A defined by aα<aβ”だね
αとβが順序数で
順序数の添え字を使って、Aに順序を導入する
順序数は、整列順序であるから
Aに整列順序が導入できた