I know that the union of countably many countable sets is countable. Is there an equivalent statement for uncountable sets, such as the union of uncountably many uncountable sets is uncountable? Furthermore, how does this generalize to other cardinals?
Union of Uncountably Many Uncountable Sets
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set-theory
cardinals
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7Since a union of uncountable sets contains an uncountable set, it is definitely uncountable. The disjoint union of uncountably many non-empty sets of any size is likewise uncountable. – 2011-12-07
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0Sure, union of sets of cardinality at least $\kappa$ has cardinality at least $\kappa$. – 2011-12-07
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0Every set is either countable (which includes empty or finite sets as well as countably infinite sets) or uncountable. By definition, uncountable means the set is not countable. There are no other choices. So, if you take 1 or more uncountable sets, it will stay in the biggest class, uncountable. Even if you take uncountably many sets that are uncountable, there's no where above uncountable to go. Uncountable isn't a cardinal. If you want to talk about cardinals, then there are infinitely many choices, unlike what I just described. – 2011-12-07
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0Your statement is true, but you should not think of countable and uncountable as being somehow analogous. In the statement "a countable union of countable sets is countable", the restriction to a countable union is necessary, because countable sets aren't supposed to be very big. Uncountable sets are very big, so *any* union of uncountable sets will be uncountable. – 2011-12-07