[FOM] iterative conception/cumulative hierarchy

[Christopher Menzel]

Am Feb 26, 2012 um 6:01 PM schrieb Nik Weaver:
> Michael Kremer wrote:
> 
>> the van Aken idea of presuppositions is intuitive and explains why
>> foundation should hold ... the point is it explains the basic idea of
>> a cumulative hierarchy without any metaphor of sets being "formed"
> 
> and Chris Menzel wrote:
> 
>> the metaphor of set formation is cashed in terms of the idea of a set *dependending on*, or *presupposing*, its members, a relation that is reflected in the (static) structure of the hierarchy
> 
> referring to Richard Heck's comment about "the idea that sets are
> *metaphysically dependent* upon their members".
> 
> So there are two distinct questions here. The first is how we are to
> repair our naive ideas about sets (viz., sets are extensions of concepts
> and every concept has an extension) in the face of the paradoxes.  The
> second is to what extent our modified understanding will support the
> ZFC axioms.
> 
> Regarding the second question, van Aken's paper is admirably frank
> about the difficulty of formulating a justification of exactly the
> right strength --- one that would legitimize power sets, for instance,
> without legitimizing full comprehension.
> 
> But we seem to be making progress on the first question.  The temporal
> metaphor of formation that I objected to has been replaced by a static
> notion of metaphysical dependence/presupposition.
> 
> Now in order to block the paradoxes, we need to agree that sets cannot
> be dependent on themselves, nor can there be cycles of dependence, or,
> I suppose, descending chains of dependence.

The paradoxes arise independent of foundation. The well-founded nature of the membership relation simply gives us a picture of the universe of sets that reveals what goes awry in the paradoxes -- some conditions on sets, notably "x is non-selfmembered", apply to sets of arbitrarily high rank; hence there is no level of the hierarchy at which they jointly constitute a set. This in turn provides us with a justification for imposing restrictions on naive comprehension that block the usual arguments to contradiction.

> But without some sort of
> temporal or spatial metaphor (in terms of the elements of a set "appearing
> before it" or "lying lower in the hierarchy") the justification for this
> condition seems to me not so clear --- we are simply presented with a
> blunt assertion that there is some notion of "metaphysical dependence"
> and it is well-founded.

Perhaps, albeit one that at least has a rigorous mathematical expression in the notion of rank.

> But even granting this formulation, I am still confused.  Under what
> conditions are we to accept that a given concept has an extension?

A concept has an extension (better, perhaps: the things a concept is true of form a SET) when there is an upper bound on the ranks of the sets it is true of.

> It sounds like the condition should be something like: a concept has an
> extension if the objects falling under its hereditary closure are
> well-founded under the "metaphysical dependence" relation.  But that
> cannot be right, because the concept *set* would, apparently, satisfy
> this condition ... so there would be a set of all sets.
> 
> So, granting that there is a notion of metaphysical presupposition and
> it is well-founded, how does that help us understand which concepts have
> extensions?  

Again, it's not well-foundedness per se that does the work; it is (as I see it) the unbounded cumulative structure that emerges in a natural way from the assumption of well-foundedness.

> Specifically, how does it explain why the concept *set* fails to have an extension?

There is no upper bound on the ranks of the sets, so there is no set of all sets. :-)

Chris Menzel