Definition 1. Let $X$ be a set.
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$X$ is countable if there is an injection from $X$ to $\N$.
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$X$ is countably infinite if there is a bijection from $X$ to $\N$.
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$X$ is uncountable if it is not countable.
Examples.
- All finite sets are countable.
- $\N$ is countable, and countably infinite.
- $\R$ is uncountable.