[FOM] iterative conception/cumulative hierarchy

[Nik Weaver]

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