[FOM] iterative conception/cumulative hierarchy

MartDowd at aol.com MartDowd at aol.com
Sat Feb 25 03:17:18 EST 2012

I have read the messages of this thread, and the question below seems to be 
 the essential point.  The cumulative hierarchy understanding of sets is  
simply that V results from iterating the power set operation through Ord.   
This seems undeniable, reducing the question of  the size of V to the  
question of  the size of Ord.  By the first incompleteness theorem,  this can 
never be "finalized'', a more formal version of the claim that Cantor's  
"absolute" can never be specified.  Only "from below" descriptions of the  size can 
be given, for example in terms of closure properties.  As I have  noted in 
previous posts, a basic "post-ZFC" property is existence of a next  
inaccessible cardinal.
For a further remark, this perspective challenges a posteriori  arguments 
for existence of "ad hoc" large cardinals, such as  measurable (or even 
Ramsey) cardinals. 
In a message dated 2/24/2012 12:17:38 P.M. Pacific Standard Time,  
nweaver at math.wustl.edu writes:

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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