Uncountable set

Definition

A set $A$ is uncountable when it is not countable, that is, when there is no injective function from $A$ to $\mathbb{N}$.

Remarks

The real numbers and the power set of $\mathbb{N}$ are uncountable. With the axiom of choice, a set is uncountable exactly when its cardinality is at least $\aleph_1$, the least cardinal larger than $\aleph_0$. A superset of an uncountable set is uncountable.

Explanations

L1 human

For an uncountable set, every attempt to list its elements misses some of them. Cantor's diagonal argument shows how to find a missing element.