They do all exist -- it's just that A(beta) is larger than the cardinal beta, so there's no contradiction.
Sent: Thursday, May 03, 2018 at 9:19 PM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun@mailman.xmission.com Subject: [math-fun] Set of all sets question
We know that the Class of all sets Sets cannot be a set, because this would imply the existence of the set
X = {s in Sets | s is not a member of s},
but X belongs to X if and only if it doesn't, hence Contradiction.
But we are told that it makes perfect sense to speak of all sets no larger than some fixed cardinal number = beta.
Let the set of all sets that are no larger than the cardinal beta be denoted by A(beta):
A(beta) = {s in Sets | card(s) <= beta}.
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)?
—Dan
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