Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|cite|improve this question

2 Answers 2

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.

share|cite|improve this answer

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 $(\emptyset, \emptyset)\in P$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.