Theories with the infinitary Vopenka property

Question

A while ago 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. 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.

Answer 1

The answer to the question is yes, assuming $VP$ holds and thus, in the terminology from your earlier question, 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.