[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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