Exercises
[086] Prerequisites:[07X],[08Z],[07V].
Let \((X,≤_ X)\) be a well-ordered non-empty set. Show that if \(S⊆ X\) is an initial segment and \((X,≤_ X)\) and \((S,≤_ X)\) are equiordinate from the map \(f:S→ X\) then \(X=S\) and \(f\) is the identity.
Solution 1(Note the difference with cardinality theory: An infinite set is in one-to-one correspondence with some of its proper subsets, cf [04G] and [04M]. Moreover, if two sets have the same cardinality then there are many bijections between them.)