 It is true (and has been the object of applications in computer science) that it is useful to admit reflexive structures for some purposes. The kind of reflexivity permitted by Aczel’s theory has been useful for some such applications. However, such structures are modelled in well-founded set theory (using relations other than membership) with hardly more difficulty, and the reflexivity admitted by Aczel’s theory (or even by a more liberal theory like that of Boffa) doesn’t come near the kind of non-well-foundedness found in genuinely alternative set theories, especially those with universal set. These theories are close variants of the usual theory ZFC, caused by perturbing the last axiom to be added to this system historically (although, to be fair, the Axiom of Foundation is the one which arguably defines the unique structure which the usual set theory is about; the anti-foundation axioms thus invite us to contemplate different, even if closely related, universal structures).