[FOM] iterative conception/cumulative hierarchy

T.Forster at dpmms.cam.ac.uk T.Forster at dpmms.cam.ac.uk
Tue Feb 28 02:30:06 EST 2012

Dear Nik (and others)

Sorry! The title is ``the iterative conception of set'' and i'm sorry i 
got the page number wrong.  The .pdf is linked from my home page.  How does 
it link to the present discussiion, as you conceive it?  The point is that 
the metaphysical presupposition relation need not be $\in$.  Language is 
always evolving of course and the phrase `iterative conception of set'' 
would, were  it being coined now, be perhaps ``the recursive conception of 

    For me the key insight was that it is possible to conceive of sets 
arising iteratively/recursively (so that you get a wellfounded metaphysical 
dependence relation - dependence relations that arise in recursive 
datatypes in this way are always wellfounded) but where that dependence 
relation is not set-membership but something else. Once you realise that 
you have TWO relations to work with not ONE a lot of things become clearer


On Feb 28 2012, Nik Weaver wrote:

>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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