Well ordering of reals from linear ordering of power set of reals

Question

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.

Answer 1

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.