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This is an exercise question from Chapter 2 of A Course in Galois Theory by D.J.H. Garling:

Suppose that $(A,\leq)$ is an infinite well-ordered set. Show that there is a unique element $a$ such that $\{x:x is infinite, while $\{x:x is finite for each $b.

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Am I missing something here? $\mathbb{N}$ is well-ordered, but does not satisfy the described property...?

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    What does Zorn's lemma have to do with this question, and why is it in the title?2012-12-09

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Indeed there seem to have a typo there. In fact two typos.

The set $\mathbb N$, or $\omega$, is a counterexample to the first point. However if there is at least a single point $a\in A$ such that $\{b\mid b is infinite, then the conclusion follows. You can show that the order of $\{b\mid b is in fact isomorphic to $\omega$, and so it is not only a counterexample it is the only counterexample.

Similarly the part about an uncountable set also contains a mistake because $\omega_1$, the least uncountable ordinal, is defined to be the set of all countable ordinals. It can be shown that this set is uncountable, and every point has only countably many predecessors, but there is no point which has uncountably many predecessors. As before, this is the only counterexample to this statement.

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    @hwhm: Yes. If every point has only countably many predecessors then there no point which has uncountably many predecessors; the other direction is as easy.2012-12-19