[FOM] iterative conception/cumulative hierarchy

[kremer at uchicago.edu]

Jim was teacher at Pitt.  Just a couple of years older than me. His early death was a tragic loss.

Michael

---- Original message ----
>Date: Fri, 24 Feb 2012 22:40:36 +0100
>From: fom-bounces at cs.nyu.edu (on behalf of Christopher Menzel <cmenzel at tamu.edu>)
>Subject: Re: [FOM] iterative conception/cumulative hierarchy  
>To: Foundations of Mathematics <fom at cs.nyu.edu>
>
>   Am Feb 24, 2012 um 9:36 PM schrieb
>   <kremer at uchicago.edu>:
>
>     Here's an old paper by Jim van Aken (RIP) which
>     explains the axioms of ZFC in terms of the idea of
>     one entity presupposing others for its existence
>     (so doing away with the notion of "forming sets"
>     from the get-go).
>
>     http://www.jstor.org/stable/2273911
>
>     Michael Kremer
>
>   Yes, good call, Michael, this is a really nice
>   paper. Along the same "stage theoretic" lines are of
>   course the classic papers by Boolos*  and Scott**
>   that Van Aken references as well as the excellent
>   2004 OUP book Set Theory and Its Philosophy by
>   Michael Potter.
>   Chris Menzel
>   *"The Iterative Conception of Set", Journal of
>   Philosophy 68 (1971), 215-231
>   **"Axiomatizing Set Theory", in T. Jech (ed)
>   Axiomatic Set Theory II, Proc. of Symp. of Pure Math
>   13, AMS, 207-214.
>
>     ---- 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_a =
>         U{P(V_b) | b < a}. 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?
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