  Finally, even if we do accept an algebraic conception of set theory—perhaps because we have a larger commitment to some kind of structuralist philosophy of mathematics—it's unclear why this commitment requires us to limit ourselves to first-order axiomatizations of set theory. After all, many of the most successful instances of the algebraic approach to axiomatization—e.g., the 19th-century results that arithmetic and analysis can be given categorical axiomatizations—turn on using a second-order background logic. And, as we noted in section 2, second-order versions of ZFC do not give rise to Skolem's Paradox. Hence, it's not enough for Skolemites to defend an algebraic approach to the axiomatization of set theory, they need to show that a first-order algebraic approach is the right way to go. See Hart 1970 and Shapiro 1991 for developments of this line of argument.