On Thu, May 3, 2018 at 10:19 PM, Dan Asimov <dasimov@earthlink.net> wrote:
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.
My intuitions go the other way, focussing on the word "bunch". A union of a *set* of other sets exists as a set. A union of a *class* of sets exists as a class. My intuition of sets is that we start from the empty set, then take the power set, then add in the power set of that, and so forth, and iterate transfinitely (at successor ordinals, add in the power set of what you've got so far; at limit ordinals, take the union of everything at smaller ordinals). Every set is associated with an ordinal, it's "birthday" (this is a theorem in ZF). The "union of everything" would have to have the largest ordinal as its birthday, and there is no largest ordinal, so it never gets born. Andy
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. ...
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