Gabriel Uzquiano has offered an account of indefinite extensibility for sets in the context of a modal logic. The modal operators are interpreted in terms of linguistic extensibility. After reviewing the proposal, I argue that the view should be understood as a version of in rebus structuralism about set theory. As such it is subject to the usual problems for in rebus structuralism. In particular, there is no good extra set-theoretic reason to assent to an ontology of sufficient cardinality to make true the theorems of ZFC.
"A Note on Gabriel Uzquiano’s “Varieties of Indefinite Extensibility”." Notre Dame J. Formal Logic 59 (3) 455 - 459, 2018. https://doi.org/10.1215/00294527-2018-0005