Some details on the relation between 2nd Order Logic and Set Theory

For Humanities researchers, who relate sentences to sentences, 2nd Order Logic seems more unavoidable than perhaps to knowledge engineers, who prefer to remain decidable. This tension forces me to revisit the precise ways in which 2nd order and first-order formalisms influence expressivity.

Now I found a nice quote in the Stanfard Encyclopedia Of Philosophy article on Higher-Order Logic about the relationship between 2nd-order logic and set-theory that illuminates some of the puzzlement about that connection:

Perhaps this [result of comparing the model hierarchy and the Levy hierarchy, RCK] would not be so puzzling if we thought of set theory as a very high order logic over the singleton of the empty set. After all, set theory permits endless iterations of the power set operation, while second-order logic permits only one iteration.