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

Dave Marker marker at math.uic.edu
Wed Oct 8 09:55:37 EDT 2008


> 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.

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