[FOM] iterative conception/cumulative hierarchy

Nik Weaver nweaver at math.wustl.edu
Mon Feb 27 22:56:56 EST 2012

Thomas Forster wrote:

> Sorry to interrupt again, but this is simply not true.   There are
> ways of blocking paradoxes other than by making $\in$ wellfounded,
> and some of these methods retain a recursive construction of V.
> If you don't want to read my article, perhaps you'd rather learn
> it from Church, whose set theory with a universal set of 1974
> allows a recursive view of V.

Thomas --- next time please include the title of your article; I
had a little trouble locating it.  (I don't have institutional access
to Rev. Symb. Logic, and your citation was slightly off --- your paper
appears on page 97, not page 1.)

I am a little stuck on how to locate your suggestion in the context of
the current discussion, as the main issue we've been debating is what
is really meant by talk about "forming sets in stages", and the general
feeling, as I have understood it, is that this kind of language is not
meant seriously, that the central idea of the iterative conception is
actually not iteration at all but rather some notion of "metaphysical

Your paper, in contrast, pushes the idea of recursion: "... although
philosophers of mathematics write of the *iterative* conception of set,
what they really mean - in the terminology of modern computer science
at least - is the *recursive* conception of sets."  So that looks to
me like a very different idea of what is really meant.  And you have
some very colorful language about this: "In the cumulative hierarchy
story, we lasso collections of sets, and then - before throwing them
back into the herd of sets whence we plucked them - we perform some
magic on the lasso contents ... The magic is performed with the aid
of a *wand*."  That's delightful imagery, but again, it's hard for me
to square it with "metaphysical presupposition".


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