I am reading the proof of Well-Ordering Theorem from wikipedia and it starts as below:
"The well-ordering theorem follows from Zorn's Lemma. Take the set A of all well-orderings of subsets of X: an element of A is an ordered pair (a,b) where a is a subset of X and b is a well-ordering of a. A can be partially ordered by continuation"
My question is, what is the well-ordering(s) which I make is as a bold above?
I thought well-ordering is a characteristic of a set, but it expresses like a noun or well-defined conceptual noun.
A well-ordering in this case is a particular ordering of a set that exhibits the well-ordering property. For example, the naturals are well-ordered because there exists an ordering of the set which is a well-ordering, simply the natural ordering, $1<2<3...$, but that's just one particular well-ordering. An equally valid ordering can be made by defining all of the odds less than the evens, $1<3<5<...<2<4<6...$