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 $M\models\mathsf{ZFC}$:



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> 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 $\omega\_1^L<\omega\_1$.
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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.



Heights of well-founded parts of models of $\mathsf{ZFC}$
The answer to the question for ordinals of uncountable cofinality is provided by the following theorem, established by Magidor, Stavi, and Shelah, in their paper [*On the standard part of nonstandard models of set theory*](https://www.jstor.org/stable/2273317), **J. Symbolic Logic** 48 (1983), no. 1, 33–38.


Note that the ordinal defined in the question as $\mathsf{wfh}(M)$ (well-founded height of $M$) is referred to below as the *standard part* of $M$.


**Theorem.** *For an ordinal* $\alpha$ *of uncountable cofinality the following are equivalent*:


**(A)** $\alpha$ *is the standard part of a nonstandard model of* ZF.


**(B)** $\alpha$ *is the standard part of a nonstandard model of* KP.


**(C)** *There exists* $\gamma > \alpha$ *such that* $\gamma$ *is a limit of ordinals which are models of* ZF, *and* $\alpha$ *has the tree property* in $L\_{\gamma}$.


**(D)** *There exists* $\gamma > \alpha$ *such that* $\gamma$ *is a limit of admissible ordinals and has the* $\alpha$ *has the tree property in* $L\_{\gamma}$.


Regarding the case when $\alpha$ has countable cofinality: in the introduction to the aforementioned paper, the authors write the following (in what follows, Friedman's theorem refers to the fact that countable admissible ordinals coincide with the heights of well-founded parts of countable models of ZF).



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> Friedman's theorem can be generalized to other ordinals, and the proof imitated, using some decomposition of $\alpha$ into small sets. (See [4].) All these generalizations handle just $\alpha$'s of cofinality $\omega$.
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In the above [4] refers to an abstract (published in the Notices of AMS, 1980), which later blossomed into another paper of Magidor, Shelah, and Stavi:
[*Countably decomposable admissible sets*](https://www.sciencedirect.com/science/article/pii/016800728490006X), **Ann. Pure Appl. Logic** 26 (1984), no. 3, 287–361.


