[FOM] iterative conception/cumulative hierarchy

Nik Weaver nweaver at math.wustl.edu
Sun Feb 26 12:01:03 EST 2012

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metaphor of formation that I objected to has been replaced by a static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.  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.

But even granting this formulation, I am still confused.  Under what
conditions are we to accept that a given concept has an extension?  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?  Specifically, how does it explain why the concept *set*
fails to have an extension?

Needless to say, my feeling is that this whole approach to the paradoxes
is completely wrong.  I am certainly open to being persuaded otherwise,
of course.


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