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I am little confused about the proof given here

On the second page, when defining $P$, the author says that $B\subset A$ and $(B,<)$ is a well-ordering structure. Isn't this exactly what we want to prove? How do we know that $B$ can be well-ordered? What happens if $P$ is empty?

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up vote 1 down vote accepted

If $A$ is non-empty then every finite subset is well-orderable, by definition of finite. So $P$ is never empty if $A$ is non-empty.

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No, the author doesn't assume what he want to prove, we just looks at the set $P$ of all $(B, <)$, where $B$ is any subset of $A$ and $<$ is a well-order on it. There are such pairs, for sure, as for example $(\emptyset, \emptyset)\in P$.

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