Independent sets cannot be equivalent if one is finite and the other infinite

Question

I am trying to show that if $\Sigma,\,\Gamma$ are independent, equivalent sets of propositions, then they are either both finite or both infinite. I'm very stuck on where to start: if $\Sigma$ finite but $\Gamma$ infinite, I think a good plan of action would be showing that $\Gamma$ cannot be independent. The only thing I can see is that for infinitely many $\alpha\in \Gamma$ we have $\Sigma \models \alpha$. Would appreciate some help.

Answer 1

HINT: Can you show (using compactness) that some finite $\Gamma_0\subseteq \Gamma$ already implies all of $\Sigma$? (Finite union of finite sets is finite . . .) What does that say about the rest of $\Gamma$?