Abstract

Surveying and criticising attitudes towards the role and strength of the iterative conception of set—widely seen as the justificatory basis of Zermelo-Fraenkel set theory with Choice—this paper highlights a tension in both contemporary and historic accounts of the iterative conception’s justificatory role: on the one hand its advocates wish to claim that it justifies ZFC, but on the other hand they abstain from stating whether the preconditions for such justification exists. Expanding the number of axioms that the conception is standardly charged with failing to justify, in the forms of the Emptyset and Powerset, this paper aims to extend the critique that the iterative conception does not justify ZFC to its subsystems. Exploring the historic and contemporary relationship between the iterative conception and set theory, the paper then attempts to defuse strategies that avoid the problems of intrinsic justification by weakening the iterative conception’s role to the ‘motivational’ or ‘heuristic’ by showing that what the iterative conception has been seen to motivate has changed over time. It is suggested that the conjunction of these arguments seriously weakens the programme of reasoning on an ‘atheoretic’ notion of set.