Why is $\sf MM^{++}$ the strongest forcing axiom? I’m no expert to the subject, but when searching around, I found Martin’s Maximum, the strongest forcing axiom. However, I soon came across $\sf MM^{++}$, which is a strengthening of $\sf MM$ and also claims to be the strongest forcing axiom. Is $\sf MM^{++}$ the definitive maximal forcing axiom or is in the sense that it’s the largest while retaining some nice property but theoretically a stronger axiom could be found?
If the latter, is there any research in that direction?
 A: To some extent, Martin's Maximum, or $\sf MM$, is named that way because if a forcing $\Bbb P$ is not stationary set preserving (SSP), then we can find a family of dense open sets $\{D_\alpha\mid\alpha<\omega_1\}$ which does not have a generic filter in $V$.
Then, we can slightly push this further, where $\sf MM^+$ comes from, by throwing into our mix a name for a stationary set and requiring that our $\cal D$-generic filter interprets this name as an actual stationary set (here $\cal D$ is a family of size $\aleph_1$ of dense open sets). Beyond that comes $\sf MM^{+\omega}$, which allows for countably many names for stationary sets, and finally $\sf MM^{++}$ that we all know and love, which pushes the names for stationary sets to allow $\aleph_1$ of those as well.
What about $\aleph_2$ names for stationary sets? Well, since $2^{\aleph_1}=\aleph_2$, we can diagonalise, in principle, over all of the stationary subsets, and run out of them to create.
So, in some very strong sense, $\sf MM^{++}$ is kind of a maximal forcing axiom.
What comes beyond that? Matteo Viale has some interesting works.

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*Viale, Matteo, Category forcings, $\mathsf{MM}^{+++}$, and generic absoluteness for the theory of strong forcing axioms, J. Am. Math. Soc. 29, No. 3, 675-728 (2016). ZBL1403.03108.


*Asperó, David; Viale, Matteo, Incompatible bounded category forcing axioms, J. Math. Log. 22, No. 2, Article ID 2250006, 76 p. (2022). ZBL07566937.
And while I am far from familiar with their details, I did hear opinions of several respected set theorists that argued that this approach is or isn't a good generalisation of $\sf MM^{++}$. But not being an expert on forcing axioms, I will reserve my judgement for when I become one.

Let me add that there is a lot of research towards "Higher Forcing Axioms" which are one of two kinds:

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*Fragments of $\sf PFA$, the Proper Forcing Axiom, or at the very least significant strengthenings Martin's Axiom, which are compatible with a larger continuum (at least $\aleph_3$).


*Identifying classes of forcings at uncountable levels which have a nice iteration theorem and that we can consistently have a forcing axiom for those. Shelah and others showed that this is impossible in the generality we would like. Namely, there are some very well-behaved forcings which do terrible things when we iterate them, even with reasonable supports. But we can still try and look for some types of forcings which are amenable to good iteration theorems and therefore consistent forcing axioms.
You can find a lot of information in the workshop notes, Methods in Higher Forcing Axioms, including problems, literature review, and more.
