Which logic is stronger? SOL or $\frak{L}_{\infty,\infty}$?

Question

Both second-order logic($\mathsf{SOL}$) and infinitary first-order logic $\frak{L}_{\infty,\infty}$ are proper extensions of first-order logic($\mathsf{FOL}$), that is, are extensions of a $\mathsf{FOL}$ and also strictly stronger than it in expressive power. It is natural to ask, are they comparable? If so, which one is more powerful?

Answer 1

I know it can sometimes be a bit irritating to give references to books here, when they may not be readily available. But Barwise and Feferman's Model-Theoretic Logics is now freely available from Project Euclid, and contains a wealth of information about infinitary and second-order logics. Chapter 9 of the classic book on second-order logic by Stewart Shapiro, sadly not so freely available, summarises many results (they are quite sensitive to the cardinality of the infinitary languages).