[FOM] iterative conception/cumulative hierarchy
Michael Lee Finney
michael.finney at metachaos.net
Sat Feb 25 14:52:14 EST 2012
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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