Are very large cardinal axioms like $I\_0$, $I\_1$, $I\_2$ consistent with $CH$ and $GCH$? 



Very Large Cardinal Axioms and Continuum Hypothesis
It is known in the folklore (I could never find a source for the results) that $I\_0$, $I\_1$, and $I\_2$ cardinals are all indestructible by small forcing, meaning that we can force ${\rm CH}$ over a universe with such a large cardinal without destroying it. I have the argument for $I\_0$ written up in lecture notes [here](http://boolesrings.org/victoriagitman/2012/04/18/models-of-rmzfc-that-are-not-definable-in-their-set-forcing-extensions/). It is basically Noah's argument with a few additional details. 


Hamkins showed [here](http://jdh.hamkins.org/fragilemeasurability/) that ${\rm GCH}$ can be forced over a universe with an $I\_1$-cardinal without destroying it and very recently Dimonte and Friedman [here](http://www.logic.univie.ac.at/~dimonte/Dimonte_Friedman.pdf) showed that ${\rm GCH}$ can be forced without destroying an $I\_0$, $I\_1$, or $I\_2$ cardinal. Indeed, they showed much more generally that any weakly increasing class function $F$ on the regular cardinals satisfying $\text{cf}(F(\alpha))>\alpha$ such that $F\upharpoonright\lambda$ is definable over $V\_\lambda$ (where $\lambda$ comes from the definition of the $I\_0$, $I\_1$, $I\_2$ cardinal) is consistent with such a large cardinal. Thus, almost any natually defined continuum pattern (on the regular cardinals), such as say $2^\kappa=\kappa^{++}$, is consistent with these large cardinals.


