I just want to record the Exercise I mentioned in class:
Suppose that is an infinite well-ordered set, and that Show that there is a bijection between and the disjoint union
To be explicit, I want a proof that makes no use of the axiom of choice. Also, although I am not requiring this as an exercise, recall that the point is to use this result to complete the proof of the following:
Theorem.If is a well-ordered set, then there is a well-ordered set such that is a proper initial segment of and there is no injection from into
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is an infinite well-ordered set, and that
Show that there is a bijection between
and the disjoint union
such that
into
A kind of library 