A well order on a subset of a power set I'm struggling with one problem which touches upon ordering of a power set. 
Assume I have a well order $\langle X,\preceq\rangle$ where $|X|=|\mathbb{R}|$. Is it possible to find (i.e. give a construction) a well ordering on an uncountable subset of $\langle P(X),\subseteq\rangle$? There is a well order there, that I know, but is there a way to explicitly show where it is? We might as well use $\langle P(\mathbb{R}),\subseteq\rangle$, but I can't come up with any well order there. 
Perhaps you've got some hints?

Every partial order embeds into its power set. Moreover this embedding is only cardinality dependent. So if $|A|\leq|B|$, every partial order of $A$ embeds into $\langle\mathcal P(B),\subseteq\rangle$.
To see this, if $\langle A,\leq\rangle$ is a partial order, $a\mapsto\{a'\in A\mid a'\leq a\}$ is an order embedding.
So yes, if $X$ is an uncountable well-ordered set, it embeds into its power set. The assumption that $|X|=|\Bbb R|$ is entirely irrelevant here.
