Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell’s “many-topoi” view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out relative to such domains; puzzles about “large categories ” and “proper classes ” are handled in a

this paper we provide consequences from and consistency results about completeness. Throughout, will denote an uncountable cardinal, and by an ideal over we shall mean a proper, -complete ideal on P() containing all singletons. If is a measurable cardinal and I a prime ideal over , then of course P()=I is complete, being the two-element Boolean algebra. The following theorem shows that completeness in itself has strong consistency strength: Theorem A. If ! 3 and there is an ideal I over such that P()=I is complete, then there is an inner model with a measurable cardinal.