[FOM] iterative conception/cumulative hierarchy

Christopher Menzel cmenzel at tamu.edu
Fri Feb 24 16:40:36 EST 2012

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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