[A while ago](https://mathoverflow.net/questions/138870/vopenkas-principle-for-non-first-order-logics) I asked about versions of Vopenka's principle for logics other than first order. Unfortunately, there doesn't seem to be much there; most logics yield the same notion, while "big" logics like $\mathcal{L}\_{\infty\omega}$ yield versions of VP which are easily disprovable.


However, going back I noticed that there is still some potentially interesting stuff around the $\mathcal{L}\_{\infty\omega}$ stuff. Say that a class $\mathbb{K}$ of structures has the *weak infinitary VP property* if, whenever $C\subseteq \mathbb{K}$ is a proper class, there are distinct $A, B\in C$ with $A\equiv\_{\infty\omega}B$. The *strong infinitary VP property* is the same, except that we ask instead for a $j: A\rightarrow B$ which is an $\mathcal{L}\_{\infty\omega}$-elementary embedding, instead of merely $\equiv\_{\infty\omega}$. Obviously the strong version is more in the spirit of the original VP; however, the weak version looks interesting too (and more manageable - see below).


Now, it's easy to come up with an example of a class *without* the weak infinitary VP property, via the following fact:


$$\mbox{$A\equiv\_{\infty\omega}B$ iff $A\cong B$ in some forcing extension.}$$


Since forcing preserves well-foundedness, no non-isomorphic ordinals can become isomorphic after forcing, so $ON$ does not have the weak infinitary VP property.


However, we can also cook up examples which *do* have it! Let's restrict attention to $\mathbb{K}$ of the form $\{M: M\models T\}$ for some complete first-order theory $T$ in a countable language with no finite models. Then:


* Since $\aleph\_0$-categoricity is $\Pi^1\_2$, it's [absolute under forcing](https://en.wikipedia.org/wiki/Absoluteness#Shoenfield.27s_absoluteness_theorem). Letting $A, B\in\mathbb{K}$ be of any cardinality, they are therefore isomorphic in any forcing extension where they are both countable - hence $A\equiv\_{\infty\omega}B$. So $\mathbb{K}$ has the weak infinitary VP.
* Similarly, $\aleph\_1$-categorical theories have the weak infinitary VP. The proof is the same, except now the claim that $\aleph\_1$-categoricity is absolute is less obvious. On the face of it, "$T$ is $\aleph\_1$-categorical" is difficult to express; however, it's equivalent to $T$ being $\omega$-stable and having no Vaughtian pairs, which is absolute.


My question is whether there are any more interesting examples of classes with the weak infinitary VP property, or *any* interesting examples of classes with the strong infinitary VP property. In particular:



> 
> **Question.** Is there an unstable theory with the strong infinitary VP property?
> 
> 
> 


*Note that one thing making the strong infinitary VP property more complicated is that $\mathcal{L}\_{\infty\omega}$-elementary embeddability doesn't seem to have as nice a characterization as $\mathcal{L}\_{\infty\omega}$-elementary equivalence - at least, none that I know of.*



Theories with the infinitary Vopenka property
The answer to the question is yes, assuming $VP$ holds and thus, in the terminology [from your earlier question](https://mathoverflow.net/questions/138870/vopenkas-principle-for-non-first-order-logics), that $VP(\mathcal{L}\_{\omega\_1 \omega})$ holds. Namely let $T$ be any unstable theory with countably many countable models; then in particular, given any $M \models T$ and any tuple $\overline{a} \in M$, then $(M, \overline{a}) \equiv\_{\infty \omega} (N, \overline{b})$ for some countable $N \models T$; thus $L\_{\infty \omega}$ embedding for models of $T$ is the same as $L\_{\omega\_1 \omega}$-embeddings.


Also, if the isomorphism relation on countable models of $T$ is Borel (in the sense of Borel complexity theory) then $L\_{\infty \omega}$-embedding on models of $T$ is the same as $L\_{\beth\_{\omega\_1}, \omega}$-embedding on models of $T$, and so if $VP(\mathcal{L}\_{\beth\_{\omega\_1}, \omega})$ holds then $T$ has the strong infinitary VP property.


