Given a first-order structure $\mathfrak{A}$ and a first-order theory $T$ one can ask if


$$
\varphi(\mathfrak{A}, T) := ``\text{there is a substructure } \mathfrak{B} \text{ of } \mathfrak{A} \text{ such that } \mathfrak{B} \models T \text{''}
$$


holds. I am interested in whether $\varphi$ is absolute between inner and outer models. More formally, let $\mathfrak{A}$, $T$, $\mathbf{V}$, $\mathbf{W}$ be such that


1. $\mathbf{V}, \mathbf{W} \models \mathsf{ZFC}$,
2. $\mathfrak{A}$ is a first-order structure and $T$ a first-order theory in $\mathbf{V}$,
3. $\mathbf{V}$ is an inner model of $\mathbf{W}$, and
4. $\mathbf{V} \models \neg \varphi(\mathfrak{A}, T)$.


Must it be the case that $\mathbf{W} \models \neg\varphi(\mathfrak{A}, T)$?



Is the existence of substructures satisfying a theory absolute?
**Assuming $T$ is countable** (in $V$), the answer is yes.


By downward Lowenheim-Skolem applied to $\mathfrak{B}$, $\varphi(\mathfrak{A},T)$ is equivalent to "$\mathfrak{A}$ has a *countable* substructure satisfying $T$." This lets us interpret $\varphi(\mathfrak{A},T)$ as asking about the ill-foundedness of a certain tree $S$; intuitively, a height-$n$ node on $S$ consists of $(i)$ a choice of the first $n$ elements of a substructure of $\mathfrak{A}$ which will satisfy $T$ and $(ii)$ some data telling us how that structure will satisfy $T$. By Mostowski absoluteness, $S$ is ill-founded in $W$ iff $S$ is ill-founded in $V$.


Countability of $T$ got used twice in the above argument; most obviously when we applied downward Lowenheim-Skolem at the start, but also in the breezed-through bulletpoint $(ii)$. If $T$ is uncountable, both these steps break down.




---


**What if $T$ is uncountable in $V$?**


*EDIT: Below I construct a counterexample where $W$ and $V$ have different cardinals. [Joel's answer](https://mathoverflow.net/a/453887/8133) shows that with more care we can avoid this, and additionally that it's the uncountability of the language as opposed to any of the structures involved that is crucial.*


Consider the language $\{c\_i:i\in\omega\}\cup\{d\_i: i\in\omega\_1^V\}\cup\{E\}$ where the $c$s and $d$s are constants and $E$ is a ternary relation symbol, and let $\mathfrak{A}$ be the two-sorted structure in $V$ consisting of $(i)$ a "point" for each $c\_i$ and $d\_j$ and $(ii)$ a "line" $l\_{i,j}$ for each pair $i\in\omega,j\in\omega\_1^V$ with $E^\mathfrak{A}=\{(c\_i,d\_j,l\_{i,j}): i\in\omega,j\in\omega\_1^V\}.$ Basically, we're looking at $K\_{\omega,\omega\_1}$ with edges being given by objects, not just relations. Note that any substructure contains all the $c\_i$s and $d\_j$s.


Now let $T$ be the theory saying that the "lines" provide a bijection between the $c$s and $d$s. This is realized in a substructure of $\mathfrak{A}$ after forcing with $Col(\omega,\omega\_1^V)$ over $V$, but isn't realized in $V$ itself.


