Minimal models in strong set theories

Question

Given some theory $T$, let $M(T)$ denote the height of the minimal model of $T$, i.e. $\min\{\eta: L_\eta \vDash T\}$. Obviously there are some famous examples, e.g. $M(\textsf{KP}) = \omega$, and $M(\textsf{KPi})$ is the least recursively inaccessible. Now, according to "2.24, A zoo of ordinals", $M(\textsf{ZFC})$ is less than the least stable ordinal ($\min\{\eta: L_\eta \prec_{\Sigma_1} L\}$). I assume this is probably true. What happens if we consider stronger theories, e.g. ZFC + there exists an I0 cardinal, and are their $M$'s still less than the least stable ordinal? Or is there some large cardinal assumption we can add to ZFC that brings the $M$ of the theory to be $\geq$ the least stable ordinal, and if so, can we find what is it?

Answer 1

Nothing like that can happen.

Note that as long as $T$ is reasonably simple (e.g. computably axiomatizable), the statement "There is a level of $L$ satisfying $T$" is $\Sigma_1$ and so reflected along $\prec_{\Sigma_1}$. This means that for such a $T$, if in fact any levels of the $L$-hierarchy satisfy $T$, there must be such a level below the least stable ordinal: if $L_\eta\prec_{\Sigma_1}L$, then since $L\models$ "There is a level of $L$ satisfying $T$" we have $L_\eta\models$ "There is a level of $L$ satisfying $T$," and so some $\alpha<\eta$ must have $L_\alpha\models T$.