[FOM] iterative conception/cumulative hierarchy
Michael Lee Finney
michael.finney at metachaos.net
Sun Feb 26 00:57:48 EST 2012
I agree with that.
While I embrace hypersets because they make so much sense, I am far
from convinced of the various extensionality axioms other than the
normal axiom -- which, of course, gives the richest set universe.
Your example is directly to the point. I do not see any reason that
there is a single set which is a member of itself. If the sets are
defined with differing predicates, it is not necessarily the case that
those predicates are equivalent which would be implied if the sets
were considered equal.
Perhaps the strong forms of extensionality are better suited to
building equivalence categories than in collapsing such sets.
Perhaps such sets should only be collapsed if the defining predicates
can be proven equivalent.
Is collapsing such sets useful outside of identifying sets with
equivalent predicates?
Michael Lee Finney
michael.finney at metachaos.net
kue> Well, but the issue of the appropriate form of the
kue> extensionality axiom is a pretty hard nut to crack with
kue> non-well-founded sets, whereas with well-founded sets it is,
kue> well, intuitively well-founded.
kue> I mean: as I recall the first formulation of the
kue> extensionality axiom for non-well-founded sets that I encountered
kue> simply posited that there is exactly one set that is its one and
kue> only member. I can see zippo justification for that (that is not
kue> circular -- maybe I have to get over my fear of circles, or
kue> something), or for any other extensionality axiom. It all looks
kue> like the realm of arbitrary choices to me. (I am open to being
kue> corrected on this. I don't claim to have kept up.)
kue> On the other hand, the van Aken idea of presuppositions is
kue> intuitive and explains why foundation should hold. (I shouldn't
kue> have said that van Aken's paper explains ZFC. I hadn't read it
kue> myself in a long time. But the point is it explains the basic
kue> idea of a cumulative hierarchy without any metaphor of sets being
kue> "formed," by replacing this with the idea of existential
kue> presupposition or dependence. At least as I recall. Of course
kue> to get to ZFC you need to add more.)
kue> Michael Kremer
kue> ---- 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
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