TITLE: Can we always "locate" a transitive model inside some larger transitive model?
QUESTION [11 upvotes]: We work in $\mathsf{ZFC}$ + "Every set is an element of a transitive model of $\mathsf{ZFC}$."

Suppose $M$ is a transitive set model of $\mathsf{ZFC}$. Say that $M$ is locatable iff there is some transitive set $N$ such that:

$N\models\mathsf{ZFC}$,

$M\in N$, and

$M$ is parameter-freely-definable in $N$.


The requirement that $N\models\mathsf{ZF}$ prevents coding $M$ into $N$ in a "silly" way. To see how this can be an issue, let $\alpha=\min\{\gamma: L_\gamma\models\mathsf{ZFC}\}$. If $N$ is a transitive model of $\mathsf{ZFC}$ with $L_\alpha\in N$, then $N$ sees that $L_\alpha$ is countable and so contains lots of (say) reals which are Cohen-generic over $L_\alpha$. Suppose $r,s$ are such reals; how is $N$ to distinguish between $L_\alpha[r]$ and $L_\alpha[s]$ in a parameter-free way? This seems to be an obstacle in general to the locatability of $L_\alpha[c]$ for $c$ Cohen-generic over $L_\alpha$. However, such an $L_\alpha[c]$ is nonetheless locatable: force over $L_\beta$, where $\beta=\min\{\gamma: \gamma>\alpha, L_\gamma\models\mathsf{ZFC}\}$, to code $c$ into the continuum pattern appropriately.
Indeed, such coding tricks give:

Every countable $M$ is locatable.

Note that the locating model is very simple - it's an element of $L(M)$. For uncountable $M$s this seems like a nontrivial further condition, and in particular none of the absoluteness theorems I'm familiar with seem to directly apply.
Similarly, we have:

If $L_\theta\models\mathsf{ZFC}$ then $L_\theta$ is locatable, even if $\theta$ is uncountable.

(For example, every ordinal $\theta$ is parameter-freely-definable in the $(\vert\theta\vert^++\theta+1)$th level of $L$ satisfying $\mathsf{ZFC}$ since $(i)$ that level of $L$ can "count" the $\mathsf{ZFC}$-satisfying levels below it, and $(ii)$ we can recover $\theta$ from $\vert\theta\vert^++\theta$ as there is a unique way to write the latter in the form $\kappa+\gamma$ for $\kappa$ a cardinal and $\gamma<\kappa$.)
However, the general situation is not clear to me:

Question: is every transitive set model of $\mathsf{ZFC}$ locatable?

Of course it is quite plausible that this will depend on additional set-theoretic hypotheses; for example, I think an affirmative answer is more likely assuming $\mathsf{V=L}$ than assuming the existence of large cardinals. But I don't immediately see how to show any of this.

REPLY [6 votes]: Great question! It has a positive answer in general:
Let $M$ be a transitive set-sized model of $\mathrm{ZFC}$. We may now find some regular cardinal $\kappa$ and some $X\subseteq\kappa$ so that $M\in L_\kappa[X]$. From now on we will work in $L[X]$. Let $\beta$ be the ordinal so that $M$ is the $\beta$-th set according to $<_X=:$ the canonical wellorder on $L[X]$. If both $X$ and $\beta$ were parameter-free definable then we could define $M$ without parameters. Unfortunately, neither of this is necessarily the case so we will try to fix that. Note that $\mathrm{GCH}$ holds in $L[X]$ at and above $\kappa$ and by our assumption on $V$, there are arbitrarily large $\lambda$ so that $L_\lambda[X]\models\mathrm{ZFC}$.

Claim: There is some $\kappa<\lambda<\kappa^+$ with

$X\in L_\lambda[X]\models\mathrm{ZFC}$
$L_\lambda[X]^{<\kappa}\subseteq L_\lambda[X]$


Proof: Let $\theta>\kappa^+$ so that $L_\theta[X]\models\mathrm{ZFC}$. Construct a continuous increasing sequence $\langle X_\alpha\mid \alpha\leq\kappa\rangle$ of elementary submodels of $L_\theta[X]$ of size $\kappa$. Make sure thate $\kappa\cup\{X\}\subseteq X_0$ and that $X_\alpha^{<\kappa}\subseteq X_{\alpha+1}$ for any $\alpha<\kappa$. Then the collapse $L_\lambda[X]$ of $X_\kappa$ is as desired.$\square$
We aim to find a generic extension of $L_\lambda[X]$ (inside $L[X]$) so that $M$ is definable there w/o parameters.
In $L_\lambda[X]$, let $\mathbb P$ be the/an  ${<}\kappa$-closed Easton-style forcing that codes $\{X, \beta\}$ (observe that $\beta<\kappa<\lambda$) into the $\mathrm{GCH}$-pattern on the successor cardinals in the interval $(\aleph_\kappa, \aleph_{\kappa+\kappa})^{L_\lambda[X]}$ and also makes $\mathrm{GCH}$ fail at $\aleph_{\kappa+\kappa+1}^{L_\lambda[X]}$. Now the following holds:

$\mathbb P$ is ${<}\kappa$-closed in $L_\lambda[X]$
$L_\lambda[X]$ has size $\kappa$
$L_\lambda[X]^{<\kappa}\subseteq L_\lambda[X]$

Thus we can construct a filter $G\subseteq\mathbb P$ (in $L[X]$!) that is generic over $L_\lambda[X]$.
Finally, $N=L_\lambda[X][G]\in L[X]\subseteq V$ is a transitive model of $\mathrm{ZFC}$ that locates $M$: First of all $\kappa$ is definable w/o parameters in $N$ since
$$N\models \aleph_{\kappa+\kappa+1}\text{ is the largest cardinal at which }\mathrm{GCH}\text{ fails}$$
Using $\kappa$, $N$ can decode $X$ and  $\beta$ from the $\mathrm{GCH}$-pattern on the successors in the interval $(\aleph_\kappa, \aleph_{\kappa+\kappa})^N$ and so can define $M$ as the $\beta$-th set according to $<_X$.