Iteration and Dependence Again

In the first part of the chapter, I clarify what is at stake in the debate between accounts of the iterative conception based on the notion of metaphysical dependence and the minimalist account I have defended in previous work (Incurvati, Philosoph Stud 159:69–87, 2012; Incurvati, Conceptions of Set and the Foundations of Mathematics, Cambridge University Press, 2020). I argue that the debate concerns how to understand and motivate the central tenet of the iterative conception that every set occurs at some level of the cumulative hierarchy. This debate, I contend, should be distinguished from the debate between actualist and potentialist accounts of the cumulative hierarchy. In the second part of the chapter, I use the distinction drawn in the first part of the chapter to assess an objection leveled by Mark Gasser (Ergo 2(1):1–26, 2015) against ante rem structuralism. In particular, this distinction makes it clear that there are two different objections in Gasser’s article. The first objection is that the iterative conception conflicts with dependence claims made by structuralists. As I have suggested in previous work, ante rem structuralists can address this objection by endorsing a minimalist account of the iterative conception. The second objection is that the indefinite extensibility of the set concept conflicts with the idea that the cumulative hierarchy is exhausted by the ZFC axioms. I chart various possible ways of addressing the second objection and show that disentangling ante rem structuralism from the idea that mathematical structures ought to be given by implicit definitions opens up a novel way of addressing the second objection. I conclude by contending that although my arguments show that ante rem structuralism is compatible with the iterative conception, there are still reasons to favour a more deflationary understanding of structuralism, advocated by John Burgess, Joel Hamkins and Gasser himself.