# [FOM] iterative conception/cumulative hierarchy

kremer at uchicago.edu kremer at uchicago.edu
Fri Feb 24 15:36:43 EST 2012

```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

---- 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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```