The philosophical problem concerning me here is nothing new. Andy has focused things by pointing to the *union* of the A(beta). It just feels intuitively that given a bunch of nested sets (that *exist*), then their union "ought to" exist as well. And the fact that in some systems the set of all sets can exist (as a class) lends credence to that intuition. —Dan ----- On Thursday, May 3, 2018, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, May 3, 2018 at 4:19 PM, Dan Asimov <dasimov@earthlink.net> wrote:
QUESTION: --------- How can it be that each A(beta) exists, and for that matter any subset of them exists, but all of them cannot exist at once (without contradiction)?
I don't understand what you think is a contradiction. All of them can and do exist at once. Their union does not exist (if using ZF) or is a class but not a set (if using Goedel-Bernay), but this is not a contradiction, because the union axiom only says that given a *set* of sets, the union set also exists. It does not guarantee the existence of the union of any class of sets exists. ...