Explanation and Plenitude in Non-Well-Founded Set Theories

Abstract Non-well-founded set theories allow set-theoretic exotica that standard ZFC will not, such as a set that has itself as its sole member. We distinguish plenitudinous non-well-founded set theories, such as Boffa set theory, that allow infinitely many such sets, from restrictive theories, such as Finsler–Aczel or AFA, that allow exactly one. Plenitudinous non-well-founded set theories face a puzzle: nothing seems to explain the identity or distinctness of various of the sets they countenance. I aim to sharpen this puzzle, make clear who it does and does not apply to, and to argue in favor of a plenitudinous theory like Boffa.