Say that an inner model $M$ of $V$ is *generically saturated* if for every forcing notion $\Bbb P\in M$, either there is an $M$-generic for $\Bbb P$ in $V$, or forcing with $\Bbb P$ over $V$ collapses cardinals.



> 
> What is the consistency strength of "$L$ is generically saturated"?
> 
> 
> 


If the answer is $0^\#$ exists, is this sort of a general answer for relative constructibility (i.e. $L[A]$ is generically saturated if and only if $A^\#$ exists)? If the answer is negative, what can we conclude from this principle?


Note, [Mohammad Golshani remarks](https://mathoverflow.net/q/207264/7206) (also [this](https://mathoverflow.net/q/184495/7206)), that assuming $0^\#$ exists for every $\kappa$, $\operatorname{Add}(\kappa,1)^L$ collapses $\kappa$ to $\omega$ (in particular, if $\kappa$ is countable in $V$, it is just a Cohen forcing). 


So in the presence of $0^\#$ at least we know that a lot of the forcings in $L$ do collapse cardinals, even if they do not admit generics in $V$ (e.g., $\operatorname{Col}(\omega,\omega\_1^V)$ cannot admit a generic, although it *does* collapse cardinals).




---


(The idea here is to marry Foreman's maximality principle that states that every forcing adds a real or collapses cardinals, with inner model hypothesis-like ideas.)



Generic saturation of inner models
The concept is inconsistent.


Consider $\mathbb{P}=Add(\omega, \kappa)\_L=Add(\omega, \kappa),$ where $\kappa$ is $(2^{\aleph\_0})^+$ of $V$. Forcing with $\mathbb{P}$ over $V$ doesn't collapse cardinals (it is ccc.c. in $V$) and there is no $M$-generic filter for $\mathbb{P}$ in $V$.


