Yeah, second-order logic is basically set theory in disguise. I'm not sure why Eliezer likes it. Example from the Wikipedia page:

There is a finite second-order theory whose only model is the real numbers if the continuum hypothesis holds and which has no model if the continuum hypothesis does not hold. This theory consists of a finite theory characterizing the real numbers as a complete Archimedean ordered field plus an axiom saying that the domain is of the first uncountable cardinality. This example illustrates that the question of whether a sentence in second-order logic is consistent is extremely subtle.