Assume $0^\#$ exists and there is an inaccessible cardinal.
Are there two transitive sets $M,N$ s.t. $M\in N,M\vDash ZF+V=L[0^\#],N\vDash ZF+V=L$?



Can local $0^\#$ exists in L?
Take a countable elementary submodel of $L\_\kappa[0^\#]$, code that into a real, note that "There is a real coding a well-founded model of $V=L[0^\#]$" is a $\Sigma^1\_2$ statement, remember what Shoenfield said about such statements: they are also true in $L$.


Now take $M$ to be a transitive collapse of some real coding that in $L$ and $N=L\_\kappa$, or some transitive collapse of a countable elementary submodel thereof.


