Existential Second Order Logic; Compactness and Löwenheim-Skolem

Question

I'm looking for proofs of Löwenheim-Skolem and Compactness in existential SoL.

I've spent a substantial amount of time on google, but can't seem to find anything!

Answer 1

This easily follows from the respective results in first order logic by skolemization.

For example let $\varphi$ be an existential second order formula of vocabulary $\sigma$: $$\varphi= \exists f_1\dots \exists f_n \exists R_1 \dots \exists R_m \psi$$

Consider a larger vocabulary $\overline{\sigma} = \sigma \cup \{f_1,\dots,f_n,R_1,\dots,R_m\}$ with new function and relation symbols.

Now let $\overline{\psi}$ be the first order $\overline{\sigma}$-formula arising by replacing all occurences of the variables $f_i$ and $R_i$ in $\psi$ with the corresponding function and relation symbols.

If $\varphi$ is satisfiable $\overline{\psi}$ is satisfiable. By the Löwenheim-Skolem-Theorem for first-order logic therefore satisfiable over a countable model. And so $\varphi$ is satisfiable over a countable model.