A concrete set of cardinality \(\aleph_1\) (Q817412)

The author constructs a set \(M\) of cardinality \(\aleph_1\) in an elementary way in order to avoid the use of stronger tools of ordinal set theory. He defines \(W\) to be the set of well-ordered subsets of the set of rational numbers \(>0\), and \(M\) as the set of equivalence classes of \(W\) where \(A\sim B\) holds for \(A,B\in W\) if there is a bijection of \(A\) onto \(B\) which preserves the \(<\)-relation.