Can the cardinal $2^{\aleph_0}$ be order-embedded in ${\cal P}(\omega)/(\text{fin})$?

Question

For $A,B\in{\cal P}(\omega)$ we say $A\subseteq^* B$ if $A\setminus B$ is finite (that is, $A$ is "almost contained" in $B$). We write $A\simeq_{\text{fin}} B$ if $A\subseteq^* B$ and $B\subseteq^* A$ (that is, the sets $A, B$ are "almost the same set" except for finitely many elements). It is easy to see that $\simeq_{\text{fin}}$ is an equivalence relation on ${\cal P}(\omega)$.

We denote the collection of equivalence classes on ${\cal P}(\omega)$ with respect to $\simeq_{\text{fin}}$ by ${\cal P}(\omega)/(\text{fin})$. Using $\subseteq^*$ on representatives of equivalence classes, it is easy to see that we can make ${\cal P}(\omega)/(\text{fin})$ into a partially ordered set.

In the Noah Schweber's beautiful post inspiring this question, we learn that there is no order-embedding from $\omega_1$ into ${\cal P}(\omega)$, but $\omega_1$ can be order-embedded in ${\cal P}(\omega)/(\text{fin})$. So within the Continuum Hypothesis ${\sf (CH)}$ we get that $2^{\aleph_0} = \omega_1$ can be order-embedded in ${\cal P}(\omega)/(\text{fin})$.

Question. Can the cardinal $2^{\aleph_0}$ (well-ordered by $\in$) be order-embedded in ${\cal P}(\omega)/(\text{fin})$ even if $\neg{\sf(CH)}$? If not: can every member of $2^{\aleph_0}$ be order-embedded in ${\cal P}(\omega)/(\text{fin})$?

Answer 1

Yes! The fact that this is consistent is originally due to Laver. Later, Baumgartner, Frankiewicz, and Zbierski strengthened Laver's result to the following theorem:

Theorem: Is it consistent that $\mathsf{MA}_{\sigma\text{-linked}}$ holds and that every Boolean algebra of size $\mathfrak{c}$ can be order-embedded in $\mathcal P(\omega)/\mathrm{fin}$.

The theorem is proved in

Baumgartner, J.; Frankiewicz, R.; Zbierski, P., Embedding of Boolean algebras in P((\omega) )/fin, Fundam. Math. 136, No. 3, 187-192 (1990). ZBL0718.03039.

On the other hand, Kunen proved in his thesis that in the Cohen model, there is no order-preserving embedding of $\omega_2$ into $\mathcal P(\omega)/\mathrm{fin}$. So the statement "every linear order of size $\mathfrak{c}$ can be order-embedded into $\mathcal P(\omega)/\mathrm{fin}$" is independent of ZFC.

I believe it is still an open problem whether the "$\mathsf{MA}_{\sigma\text{-linked}}$" in the theorem above can be strengthened to simply "MA". I seem to remember hearing at one point that Woodin had proved something about this, but I'm not sure it was ever published and I don't remember exactly what he proved.