[FOM] large chains in the power set of the reals

[Dave Marker]

>Question:

> Is the independence from ZFC of the following statement known?

> "There are chains in the power set of reals under inclusion, which have
> the same cardinality as the power set of reals."

For any infinite cardinal kappa there is a chain of length 2^kappa
in the power set of kappa.  Let (D,<) be a dense linear order
of cardinality kappa with 2^kappa cuts.  The chain of lower cuts
in D has length 2^kappa.