> [!NOTE] **Definition (Finite set)** > A [[Sets|set]] $S$ is finite if there exists a [[Bijection|bijection]] between $S$ and $\{ i \in \mathbb{N} \mid i< n \}$ for some $n \in \mathbb{N}.$ **Note**: Equivalently, $S$ is finite iff there is a surjective [[List|list]] of some length whose elements are taken from $S.$ In this case, the [[Cardinality|cardinality]] of the set is the length of the list. A set that is not finite is infinite: that is, it does not have the same cardinality as $\{ i \in \mathbb{N} \mid i< n \}$ for any $n \in \mathbb{N} \cup \{ 0 \}$. # Properties **Pigeonhole principle**: The [[Pigeonhole Principle|pigeonhole principle]] asserts that there is no injection between finite sets $A$ and $B$ if $|A|>| B|.$ **Cardinality is well defined**: In the definition above, note that $n$ is a [[Cardinality|cardinality]] of $S.$ It is not possible that a finite set has the same cardinality as $[\![n]\!]$ for more than one $n\in \mathbb{N}$ by [[Uniqueness of cardinality of finite sets|uniqueness of cardinality of finite sets]].