[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