# Chains of length \$2^kappa\$ in \${cal P}(kappa)\$

It is a fact that continues to boggle my mind: There is a set $${cal C}subseteq {cal P}(omega)$$ such that $$|{cal C}|=frak{c}=2^{aleph_0}$$ and for all $$A,Bin{cal C}$$ we have $$Asubseteq B$$ or $$Bsubseteq A$$ for all $$A,Bin {cal C}$$. (In that case we call $${cal C}$$ a chain.)

The proof is remarkably easy: pick any bijection $$varphi:omegatomathbb{Q}$$ and consider the collection $${cal D}$$ of all down-sets of $$mathbb{Q}$$. This collection forms the Dedekind cuts which can be used to construct $$mathbb{R}$$, which has cardinality $$frak{c}$$. Now $${varphi^{-1}(D):Din{cal D}}$$ is a chain in $${cal P}(omega)$$ of cardinality $$frak{c}$$.

Now you would think that you could carry the construction of $$mathbb{Q}$$ and Dedekind cuts to cardinals larger than $$omega$$ — but I was not able to.

Question. If $$kappa$$ is an infinite cardinal, is there a chain $${cal C}subseteq {cal P}(kappa)$$ such that $$|{cal C}|=2^kappa$$?