Heights of well-founded parts of models of $\mathsf{ZFC}$

Question

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.

Answer 1

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, 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 $\alpha$ 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).

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 $\omega$.

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, Ann. Pure Appl. Logic 26 (1984), no. 3, 287–361.