Toward a modal-structural interpretation of set theory (Q1202500)

Elsewhere [the author, Mathematics without numbers. Towards a modal- structural interpretation (1989; Zbl 0688.03001)] we have indicated how a framework of second-order modal logic can serve to represent a structuralist interpretation of basic mathematical theories such as number theory (PA) and real analysis (RA). On the interpretation in question (which we abbreviate ``msi'', for ``modal-structural interpretation''), ordinary mathematical statements are construed as elliptical for hypothetical statements as to what would hold in any structure of the appropriate type, this being describable directly in second-order logical notation (using a sufficient, finite set of axioms, suitably relativized, and generalizing on the relational primitives of the theory). Absolute reference to mathematical objects is eliminated entirely. Instead, there is, in addition to the translation scheme (which we call the ``hypothetical component''), a categorical component to the effect that structures of the appropriate type are logically possible. This was found to be an irreducible, nonanalytic working hypothesis associated with the mathematical practice codified in the theory in question \((\text{PA}^ 2,\text{RA}^ 2\), etc.). It was then shown how both the traditional `objects platonist' and the modal-struturalist, working from within their respective frameworks, can justify the interpretation as accurate and adequate. In particular, the modal- structuralist can establish the truth-determinateness of the translation scheme by recovering suitable versions of the categoricity of the theories in question, and the `isomorphism theorem' (that isomorphic structures satisfy the same sentences). Here we confront the task of extending this sort of interpretation to set theory itself.