[FOM] iterative conception/cumulative hierarchy
kremer at uchicago.edu
kremer at uchicago.edu
Sat Feb 25 16:34:02 EST 2012
Well, but the issue of the appropriate form of the extensionality axiom is a pretty hard nut to crack with non-well-founded sets, whereas with well-founded sets it is, well, intuitively well-founded.
I mean: as I recall the first formulation of the extensionality axiom for non-well-founded sets that I encountered simply posited that there is exactly one set that is its one and only member. I can see zippo justification for that (that is not circular -- maybe I have to get over my fear of circles, or something), or for any other extensionality axiom. It all looks like the realm of arbitrary choices to me. (I am open to being corrected on this. I don't claim to have kept up.)
On the other hand, the van Aken idea of presuppositions is intuitive and explains why foundation should hold. (I shouldn't have said that van Aken's paper explains ZFC. I hadn't read it myself in a long time. But the point is it explains the basic idea of a cumulative hierarchy without any metaphor of sets being "formed," by replacing this with the idea of existential presupposition or dependence. At least as I recall. Of course to get to ZFC you need to add more.)
Michael Kremer
---- Original message ----
>Date: Sat, 25 Feb 2012 14:52:14 -0500
>From: fom-bounces at cs.nyu.edu (on behalf of Michael Lee Finney <michael.finney at metachaos.net>)
>Subject: Re: [FOM] iterative conception/cumulative hierarchy
>To: Foundations of Mathematics <fom at cs.nyu.edu>
>
>Thank you.
>
>I have seen that paper before, but its been a while. I downloaded it
>again (another $10, arrggghhh!) and reread it.
>
>However, that doesn't change my opinion. Essentially, I reject both
>the doctrine of limitation of size and the doctrine of the hierarchy.
>Nor do I consider it necessary to reject inconsistency which desire
>originally motivated both.
>
>I consider these to be emotional crutches that many people trained in
>the 1800's needed to deal with infinities. And, because of the paradoxes,
>they sort of stuck (both with and without the 't'). But, if we can
>change the axioms of Geometry after 1800 years, surely we can change
>the axioms of Set Theory after less than 100.
>
>A more interesting paper is "Complete Totalities" by Rafi Shalom which
>can be found (free!) at http://arxiv.org/abs/1107.3519 where the idea
>is that sets are "just there at once".
>
>Once Peter Aczel did his work on non-well-founded sets, I cannot see
>any justifiable reason to accept the axiom of regularity. The main
>question is what form should the extensionality axiom take (since there
>are competing variants)? Also, while pointed graphs may make good
>expositional tools, they have no place in the founding axioms (simply
>because they haven't been defined at that point in the axioms).
>
>
>Michael Lee Finney
>michael.finney at metachaos.net
>
>kue> Here's an old paper by Jim van Aken (RIP) which explains
>kue> the axioms of ZFC in terms of the idea of one entity presupposing
>kue> others for its existence (so doing away with the notion of
>kue> "forming sets" from the get-go).
>
>kue> http://www.jstor.org/stable/2273911
>
>kue> Michael Kremer
>
>kue> ---- Original message ----
>>>Date: Thu, 23 Feb 2012 08:13:32 -0600 (CST)
>>>From: fom-bounces at cs.nyu.edu (on behalf of Nik Weaver <nweaver at math.wustl.edu>)
>>>Subject: [FOM] iterative conception/cumulative hierarchy
>>>To: fom at cs.nyu.edu
>>>
>>>
>>>Chris Menzel wrote:
>>>
>>>> The metaphor of "forming" sets in successive stages that is often
>>>> invoked in informal expositions of the cumulative hierarchy is just
>>>> that, a metaphor; some people find it helpful in priming the necessary
>>>> intuitions for approaching the actual mathematics. But in ZF proper, the
>>>> metaphor is gone; there are indeed "stages", or "levels", but these are
>>>> fixed mathematical objects of the form V_? = ?{?(V_?) | ? < ?}. The
>>>> cumulative hierarchy is indeed "there all at once", just as you desire.
>>>
>>>As I understand it, the *iterative conception* is the idea that sets
>>>are formed in stages, and the *cumulative hierarchy* is the structure
>>>this imposes on the set theoretic universe. The iterative conception
>>>is universally explained in terms of "forming" sets in "stages" (often
>>>with the scare quotes included). Once the explanation is complete this
>>>language is then, universally, retracted.
>>>
>>>Is "Sets are formed in stages --- but not really" not a fair summary
>>>of the iterative conception?
>>>
>>>Without invoking the "metaphor" of formation in stages, what is the
>>>explanation of why we should understand the universe of sets to be
>>>layered in a cumulative hierarchy?
>>>
>>>Nik Weaver
>>>Math Dept.
>>>Washington University
>>>St. Louis, MO 63130
>>>nweaver at math.wustl.edu
>>>_______________________________________________
>>>FOM mailing list
>>>FOM at cs.nyu.edu
>>>http://www.cs.nyu.edu/mailman/listinfo/fom
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