Countable and uncountable sets
A set is countable when it is equinumerous with a subset of $\omega$. This includes all finite sets, including the empty set, and the infinite countable sets are said to be countably infinite. An uncountable set is a set that is not countable. The existence of uncountable sets is a consequence of Cantor's observationt that the set of reals is uncountable.
Uncountability of the reals
Cantor's diagonal argument shows that the set of reals is uncountable.
Uncountability of power sets
More generally, the power set of any set is a set of strictly larger cardinality.
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