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

[Liang Yu]

This question was circulated by Terwijn at LCR 2007 Argentina. The question
is motivated by the study in M-degree.
 
I think that it is still open. Actually I tried from time to time but fails.
What I know are:

Given a linear ordering $\mathbb{L}=(L,\leq_L)$, a subset $Q
\subseteq L$ is dense if for all $a_1, a_2 \in L$, there exists
$q\in Q$ so that $a_1 \leq_L q \leq_L a_2$.
\begin{definition}
Given cardinals $\kappa\leq \lambda$, we use
$\mathrm{T}(\kappa,\lambda)$ to denote that {\em there is a chain of
size $\lambda$ in $(\mathcal{P}(\kappa),\subset)$},
$\mathrm{M}(\kappa,\lambda)$ to denote that {\em there is a linear
ordering of cardinality $\lambda$ with a dense subset of cardinality
$\kappa$} and $\mathrm{B}(\kappa,\lambda)$ to denote that {\em there
is a $\kappa$-tree with $\lambda$-many branches}.
\end{definition}

Here a {\em $\kappa$-tree} $T$ \footnote{Note
that the definition here is different with the standard one.} is a
subset of $2^{<\kappa}$ of size $\leq \kappa$.

It is not difficult to show that  $\mathrm{T}(\kappa,\lambda)$ is equivalent
to $\mathrm{M}(\kappa,\lambda)$ for $\lambda \geq \kappa$. Moreover,
$\mathrm{T}(2^{aleph_0},2^{2^{\aleph_0}})$ is simply a theorem of ZFC+CH and
ZFC proves that $\mathrm{T}(2^{aleph_0},(2^{\aleph_0})^+)$


Baumgartner \cite{Bau70} and Todor\v{c}evic \cite{Tor80}
proved that $\neg \mathrm{B}(\aleph_1,\aleph_2)$ is consistent with
Martin's axiom under the assumption of the existence of an
inaccessible cardinal.(I didn't find Baumgartner's paper. Todor\v{c}evic's
published paper seems incorrect to me. But he has a manuscript containing a
nice and of course correct proof, a so-called mixed forcing).

The reason that the question is so difficult is related to the limitation of
proper forcing, I think.

Email me if you have any further questions.

Liang