# Countable and uncountable sets

From Cantor's Attic

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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