In this talk Barton will argue that we're now at a conceptual crossroads regarding the iterative conception of set. To do this Barton will appeal to work on conceptual engineering. Barton will argue 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. Barton will 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, Barton will 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. Barton will explain two such conceptions, and tentatively conclude that discussion 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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