Finite and Infinite Sets
This note deals with the standard definition of what it means for a set to be finite or infinite. Note that the other common definition called Dedekind finiteness is equivalent under the axiom of choice.
Definition
A set \(S\) is called finite if it is empty, or there is a natural number \(n\) for which there exists a bijection:
\[ S \to \{1, \dots, n\}.\]
By convention, we take \(\{1, \dots, n\}\) to be \(\varnothing\) when \(n = 0\), such that \(\varnothing\) is finite.
Definition
A set is called infinite if is not finite.