Jack Silver proved that if $x$ is a real so that every $x$-admissible ordinal is a cardinal in $L$, then $0^{\sharp}$ exists.


I wonder whether various weaker or stronger versions of Silver's result have been considered in the literature. For example,


$\bf{Question \ 1.}$:
How strong is the statement that there is real $x$ so that every $x$-admissible ordinal is a recursively inaccessible?


$\bf{Question \ 2.}$:
How strong is the statement that there is real $x$ so that every $x$-admissible ordinal is inaccessible in L?



Concerning Silver's result
Re Q1: Sy Friedman has shown by class forcing over L that there can be consistently a real r
with the r-admissibles precisely the recursively inaccessibles. (S Friedman, "Strong Coding" APAL, vol 35,1987).


Re Q2: All $0^\sharp$-admissibles are limits (indeed fixed points in the enumeration) of Silver indiscernibles (as can be seen by iterating the $0^\sharp$-mouse inside the least admissible set containing it). So 2 is not a strengthening.
One obtains strengthenings really by changing the model $L$ to some other inner model, such as a core model, for example $L^\mu$ the least inner model with a measurable cardinal.


