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I'm very low about set theory. Excuse the simplicity of the problem.

Can the infinite number of union of infinitely countable sets be an infinitely uncountable set?

But, Intuitively I would say:

The union of finite number of infinitely countable set, must be an infinitely countable set.

But, if the number of sets is an infinite number, I'm not sure.

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  • $\begingroup$ The union of a countable collection of countable sets (whether finite or infinite) is countable. On the other hand the union of an uncountable collection of disjoint singleton sets is uncountable (otherwise you could count the sets by identifying them with their elements). To get infinite countable sets just take the union of each singleton with the natural numbers. $\endgroup$ – Mark Bennet May 6 at 9:24
  • $\begingroup$ @MarkBennet I understood. Thank you. So, as I understand, number of union finite or infinite is not important. $\endgroup$ – Learner May 6 at 9:32
  • $\begingroup$ @MarkBennet To prove that a countable union of countable sets is countable does involve use of some form of the axiom of choice. $\endgroup$ – Lord Shark the Unknown May 6 at 10:43
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Yes, of course.

For a real number $a$, take the infinitely countable set $S_a := a + \mathbb{Z}$. Then take the union $$\bigcup_{a \in [0,1)} S_a = \mathbb{R}$$ to get an uncountable set.

However, if you take a union over a countable set, so $$\bigcup_{x \in I} M_x$$

and $I$ and all $M_x$ are countable, then this union will also be countable. A proof for that can be found, for example, here.

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  • $\begingroup$ (+) Thank you. The second part of Question can apply for the number of union is finite or infinite. Both are correct. am I right? $\endgroup$ – Learner May 6 at 9:34
  • $\begingroup$ Yes, the second part contains both finite sets and infinitely countable ones. $\endgroup$ – Dirk May 6 at 11:01

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