Exercises
[22F]Prerequisites:[21W],[071],[07V],[07X].Difficulty:*.Note:exercise 2 written exam on 29 January 2021.(Solved on 2022-10-13
in part)Let be given \((X,≤_ X )\) where \(X\) is an infinite set and \(≤_ X\) is a well ordering.
If \(X\) has no maximum, then there exists \((Y,≤_ Y )\) such that setting \(Z=Y× ℕ\) with \(≤_ Z\) the lexicographical order, then \((X,≤_ X)\) and \((Z,≤_ Z)\) have the same type of order.
If instead \(X\) has maximum, then there exist \((Y,≤_ Y )\) and \(k∈ℕ\) such that, setting \(Z\) be the concatenation of \(Y× ℕ\) and \(\{ 0,\ldots k\} \) (where \(Y×ℕ\) has the lexicographical order, as above), then \((X,≤_ X)\) and \((Z,≤_ Z)\) have the same type of order.
Show that, in the previous cases, \(Y\) is well ordered.
Solution 1