Open-endedness, schemas and ontological commitment

Second-order axiomatizations of certain important mathematical theories - such as arithmetic and real analysis - can be shown to be categorical. Categoricity implies semantic completeness, and semantic completeness in turn implies determinacy of truth-value. Second-order axiomatizations are thus appealing to realists as they sometimes seem to offer support for the realist thesis that mathematical statements have determinate truth-values. The status of second-order logic is a controversial issue, however. Worries about ontological commitment have been influential in the debate. Recently, Vann McGee has argued that one can get some of the technical advantages of second-order axiomatizations - categoricity, in particular - while walking free of worries about ontological commitment. In so arguing he appeals to the notion of an open-ended schema - a schema that holds no matter how the language of the relevant theory is extended. Contra McGee, we argue that second-order quantification and open-ended schemas are on a par when it comes to ontological commitment.