5
$\begingroup$

Working without the axiom of choice (ZF or even ZF + DC), can we show the existence of a well ordering of the real line $\mathbb{R}$ assuming that there is a linear ordering of $\mathcal{P}(\mathbb{R})$? I know that we can construct a linear ordering of $\mathcal{P}(\mathbb{R})$ using a well ordering of $\mathbb{R}$. But I don't see how to prove the converse if it is possible at all.

$\endgroup$
5
$\begingroup$

No.

It is consistent that the reals cannot be well-ordered, but every set can be linearly ordered. For example, in Cohen's first model, where the Boolean Prime Ideal theorem holds, but there is a Dedekind-finite set.

You can use Pincus' results about "Adding DC" to obtain a model where also DC holds. And probably there are other simpler proofs of this with DC too.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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