[FOM] iterative conception/cumulative hierarchy

[kremer at uchicago.edu]

Nik Weaver distinguishes two questions:

(1)  how we are to repair our naive ideas about sets ... in the face of the paradoxes.  (2)to what extent our modified understanding will support the ZFC axioms.

He points out that the idea of presupposition/metaphysical dependence is supposed to be an answer to (1) but not (a full answer) to (2).  He then makes two points.

(a)  in order to block the paradoxes, we need to agree that dependence is well-founded. This seems unjustified without reverting to a temporal or spatial metaphor.

(b) Even if metaphysical dependence is well-founded, we do not have a clear statement of conditions under which a given concept has an extension. He considers "a concept has an extension if the objects falling under its hereditary closure are well-founded under the "metaphysical dependence" relation," but claims that "the concept *set* would ... satisfy this condition." He asks how accepting a well-founded metaphysical dependence relation can "help us understand which concepts have extensions?  Specifically, how does it explain why the concept *set* fails to have an extension?"

On (a), I think the idea that dependence/presupposition is well-founded is intuitive and does not depend on spatial and temporal metaphors, but we may just have to leave matters there and agree to disagree. Perhaps others can do better.

On (b), though, I have two points.  First, I think that the presupposition idea is not giving us a sufficient condition for a concept to have an extension.  It is only giving us a necessary condition.  For this reason it cannot give us a positive criterion ("a concept has an extension if...." -- as in Weaver's attempted formulation) but only a negative criterion ("a concept fails to have an extension if..." -- or "if a concept has an extension, then...").  And that formulation is simply this:

If a concept has an extension, then that the hereditary closure of that extension is well-founded under the metaphysical dependence relation.  (I mean to be using "x is well-founded under the metaphysical dependence relation" in the same sense that Weaver uses it).

Put negatively:

If the hereditary closure of the extension of a concept would be non-well-founded under the metaphysical dependence relation were it to exist (as a set), then that concept does not have an extension.

My second point is that contrary what Weaver says, this  criterion *does* rule out the set of all sets.  The reason is simple:  if there is a set of all sets, S, then S is a member of S. And so there is an object in the hereditary closure of S, namely S itself, which is not well-founded under the metaphysical dependence relation.  (Since we are granting (a).)

Michael Kremer