From: hbe@sonia.math.ucla.edu (H. Enderton)
Subject: Re: infinitely iterated power set
Date: 15 Jan 2000 03:51:49 GMT
Newsgroups: sci.logic,sci.math
Summary: [missing]

>maky m.  <mmanch01@my-deja.com> wrote:
>>in a discussion with a colleague, the following construct arose:
>>P1(A) := power set of set A
>>Pi(A) := P(Pi-1(A)) (ie the power set of the set Pi-1(A)) for i a
>>positive integer greater than 1
>>question: is the limit of Pi(A) as i goes to infinity a set?

Yes.  But you need the axiom of replacement to prove this.
Zermelo set theory does not suffice.

Dave Seaman <ags@seaman.cc.purdue.edu> wrote:
>This process can be continued.  You can talk about the (w+1)st power set
>of A, the (w+w)th power set, and in general, the alpha-th power set for
>any ordinal alpha.  The question then arises:  does this process have a
>limit?  Is there a set big enough to contain all of these iterated power
>sets? > >The answer is no. 

Indeed, if you take A = emptyset, and iterate over all the ordinals,
you get V, the class of all sets.  That's the foundation axiom.

--Herb Enderton
  hbe@math.ucla.edu