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](https://mathoverflow.net/a/440492/8628) 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 ${\cal P}(\omega)/(\text{fin})$?



Can the cardinal $2^{\aleph_0}$ be order-embedded in ${\cal P}(\omega)/(\text{fin})$?
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**](https://doi.org/10.4064/fm-136-3-187-192), Fundam. Math. 136, No. 3, 187-192 (1990). [ZBL0718.03039](https://zbmath.org/?q=an:0718.03039).


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.


