Is there a specific infinitary sentence second-order logic can't capture?

Solution 1:

Here's a partial answer:

It's easy to show that for $X\subseteq\omega$ the (isomorphism class of) structure $$Set_X:=(\omega; <,X)$$ is characterizable by a single second-order sentence iff $X$ is second-order-definable in $(\omega;<)$ - that is, iff $X$ is a lightface projective real. However, we obviously have that $Set_X$ is characterizable by a single $\mathcal{L}_{\omega_1,\omega}$-sentence. So, for example, letting $\sigma$ be the Scott sentence of $Set_{Th_2(\omega;<)}$ we have that no second-order sentence is equivalent to $\sigma$ (even on countable structures).

However, this isn't entirely satisfying: this property of $\sigma$ may not be absolute upwards since $Th_2(\omega;<)$ is not absolute upwards in general. Specifically, while large cardinals do yield projective absoluteness, this breaks down quite badly if we work over $L$ since $Th_2(\omega;<)^L$ is second-order definable over $(\omega;<)$ in the sense of $L^G$ when $G$ is $Col(\omega_1^L,\omega)$-generic over $L$.

• The point is that - regardless of $V$ - if $\theta$ is a second-order sentence then $L\models((\omega;<)\models\theta)$ iff $L_{\omega_1^L}\models\hat{\theta}$ for an appropriate first-order sentence $\hat{\theta}$ in the language of set theory. If $\omega_1^L$ is countable, then $L_{\omega_1^L}$ is characterizable up to isomorphism as a countable well-founded structure satisfying the obvious fragment of $ZFC+V=L$ and such that there is no larger countable well-founded model of that same theory which is locally countable. For each second-order sentence $\sigma$, the sentence $\sigma' \equiv$ "every such structure thinks $\sigma$ is true" is then a second-order sentence over $(\omega;<)$. (And the maps $\theta\mapsto\hat{\theta},\sigma\mapsto\sigma'$ are simple enough that they don't cause issues.)

Indeed, it's not hard to show that there is a parameter-freely-definable set forcing in $L$ such that for every generic $G$, all constructible reals are second-order definable over $(\omega;<)$ in the sense of $L[G]$. So this solution isn't "persistent to outer models," even if we restrict attention to pretty mild constructions.