[FOM] iterative conception/cumulative hierarchy

Richard Heck rgheck at brown.edu
Fri Feb 24 16:00:27 EST 2012

On 02/23/2012 09:13 AM, Nik Weaver wrote:
> 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?
Without defending the view myself...the language of "forming" has an 
ambiguity that allows what is supposed to be a story about the *nature* 
of sets be presented instead as a story about how God (or Cantor) 
created them. The metaphor of formation is rightly withdrawn when it is 
understood in the "creation" sense; but it need not be withdrawn when 
understood in a different sense, the one for which people are reaching 
when they say such things as that a set "has its being" in its members. 
What motivates the iterative conception, so understood, is thus the idea 
that sets are *metaphysically dependent* upon their members.

One may or may not find the idea plausible, or even intelligible. But it 
does seem clear, to me at least, that the language of formation and 
stages can reasonably be understood as an attempt to give expression to 
it. If so, then the "iterative conception" doesn't have to be understood 
as a Big Lie.

Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University

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