This will be a talk on 21 June 2022 at the conference *Engineering The Concept of Collection.* The handout can be found here.

Abstract: In this talk I’ll present a main argument of a short book I’m working on entitled *Engineering Set-Theoretic Concepts* (I’m interested in comments on the draft, so please get in touch if you’d like to see it once ready). I’ll first note that conceptual engineering has formed a part of set-theoretic activity since its inception as a mainstream area of mathematical research, and that the development of the iterative (and other) conceptions of set was in part responding to inconsistency in the naive set-concept. I’ll then argue that whilst the iterative conception can be taken to be a consistent concept in its own right, it is deficient in various ways (in particular, it fails to tell us enough about the nature of infinite sets). Contemporary set theory, I’ll argue, has now moved to a maximal iterative conception of set, and this conception is inconsistent. Many contemporary accounts of the ontology underlying set-theoretic practice should be conceived of as attempts to engineer consistent conceptions of the maximal iterative concept of set. I’ll explain two such conceptions, the directed and schematic iterative conceptions of set. I’ll tentatively conclude that discussion in the philosophy of set theory should focus less on the vexed and seemingly intractable issue of ontology, and instead concern itself more with the (nonetheless difficult) question of the relative theoretical virtues of alternative conceptions.

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