Is it always true that a family of countable sets is always countable?
If I prove that every $A\in B$ is countable is it enough to prove $B$ is countable?
2026-04-18 05:01:51.1776488511
The set of countable subsets is countable
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The set of natural numbers is countable. As such, the power set of $\mathbb{N}$, consisting of all subsets of $\mathbb{N}$, is a family of countable sets. However, the power set of a set must have a larger cardinality than that set, so the power set of the naturals is uncountable. In particular, it has cardinality $2^{\beth_0}=\beth_1$.