[FOM] iterative conception/cumulative hierarchy

Nik Weaver nweaver at math.wustl.edu
Tue Feb 28 17:44:20 EST 2012

Chris Menzel wrote (quoting me):

>> 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.

Hold on ... the unbounded cumulative structure doesn't emerge from the
assumption of well-foundedness.  That came out of van Aken's paper very
clearly.  You need to separately adopt a reflection principle, one of just
the right strength, which apparently lacks a good informal justification.

Right?  We decided that the iterative conception isn't meant literally.
It's just a stand-in for some notion of "metaphysical dependence" or
"presupposition".  But when that notion is axiomatized you get an
extremely weak system (van Aken's MSU) that captures no unbounded
cumulative structure.

So it turns out that the iterative conception does very little toward
telling us what concepts have extensions.  All the work is done by a
reflection principle which has no clear justification.


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