Here $\operatorname{cof}(\mathcal{L})$ refers to the largest cardinal characteristics of the continuum in Cichon's diagram. My question is:


* Is the theory $\mathsf{ZFC}+ \operatorname{cof}(\mathcal{L})=\aleph\_1 + 2^{\aleph\_0}=\aleph\_3$ known to be consistent?



$\operatorname{cof}(\mathcal{L}) = \aleph_1$ and $2^{\aleph_0} = \aleph_3$
Yes, this is consistent. Probably the simplest way to get a model of this theory is to begin with a model of GCH, and then force with a countable support product of $\aleph\_3$ copies of the Sacks poset. This is sometimes called the "side-by-side" Sacks model, and you can get any value of $\mathfrak{c}$ you like in a model like this (as opposed to the iterated Sacks model, in which $\mathfrak{c} = \aleph\_2$). A countable support product of Sacks posets has the Sacks property, and this ensures that the measure-$0$ Borel sets coded in the ground model generate the ideal of measure-$0$ sets in the extension. This, plus the CH holding in our ground model, implies $\mathrm{cof}(\mathcal N) = \aleph\_1$ in the extension. All this stuff (along with a lot of other cool things about Sacks extensions) is proved in detail in [this](https://www.math.uni-hamburg.de/home/geschke/papers/OnSacks_main.pdf) survey paper of Geschke and Quickert.


