Work in the theory $\mathsf{ZFC}$ + "Every set is contained in some transitive model of $\mathsf{ZFC}$." My question is the following: which ordinals are the heights of the well-founded parts of models of $\mathsf{ZFC}$?

*For what follows, let $\mathsf{wfh}(M)$ denote the height of the well-founded part of $M$.*

In the **countable** case, the answer is simple: a countable ordinal $\alpha>\omega$ is the height of the well-founded part of a (WLOG countable) model of $\mathsf{ZFC}$ iff $\alpha$ is admissible. The left-to-right direction holds since the well-founded part of any admissibile set is itself admissible; the interesting direction is right-to-left, where Barwise compactness comes into play.

In the **uncountable** case the left-to-right direction of the argument above still works, but the right-to-left direction breaks since we lose Barwise compactness. In fact, it's consistent with $\mathsf{ZFC}$ that there are admissible ordinals which are not of the form $\mathsf{wfh}(M)$ for any $M\models\mathsf{ZFC}$:

Suppose $M\models\mathsf{ZFC}$ and $\mathsf{wfh}(M)=\omega_1^L$. Then we have (via mild abuse) that $L_{\omega_1^L}\subseteq M$. Since $L_{\omega_1^L}$ is locally countable, this means that $\omega_1^M$ must be ill-founded. Picking some ill-founded $M$-ordinal $\alpha<\omega_1^M$, we get in $M$ a bijection $\alpha\rightarrow\omega$ - which restricts externally to an injection $\omega_1^L\rightarrow\omega$. So we must have $\omega_1^L<\omega_1$.

More generally, whenever $\kappa$ is an infinite cardinal such that $\kappa^+=(\kappa^+)^L$ we get that $\kappa^+\not=\mathsf{wfh}(M)$ for all $M\models\mathsf{ZFC}$ - just run the argument above with $\kappa^+$ in place of $\omega_1$ and "locally size-$\le\kappa$" in place of "locally countable." But this doesn't give a $\mathsf{ZFC}$ result since it's consistent that $L$ *never* computes successor cardinals correctly. So it's not even obvious to me that $\mathsf{ZFC}$ disproves "The possible values of $\mathsf{wfh}(M)$ are exactly the admissible ordinals." In particular note that it **is** consistent that $\omega_1$ is the height of the well-founded part of a model of $\mathsf{ZFC}$, since in fact it's consistent that $L_{\omega_1}\models\mathsf{ZFC}$ outright.