This is a crosspost of this question that I posted on the math stack exchange a year and a half ago.
There are two slightly different versions of compactness in the FOL setting:
- If $\Delta$ is finitely satisfiable, then $\Delta$ is satisfiable.
- If $\Gamma \models \varphi$ then there exists a finite subset $\Gamma_0 \subset \Gamma$ such that $\Gamma_0 \models \varphi$.
I'm interested in the second one here because positive second-order logic does not have actual contradictions. For second-order logic though, these two notions line up.
Consider second-order logic with a single non-empty domain. We have function and relation symbols as vocabulary items. A constant is a nullary function.
Additionally, quantifiers can introduce functions and relations of arbitrary arity (e.g. $\exists f / 2$ or $\forall R / 3$). Suppose our connectives are $\land$, $\lor$ and $\lnot$.
Consider the fragment without $\lnot$. Let's call this positive second-order logic.
Positive second-order logic with Henkin semantics is definitely compact.
I'm wondering whether positive second-order logic with full semantics is compact.
