> [!NOTE] Definition (Set-theoretic definition of Natural Numbers) > The [[Zermelo Frankel set theory (ZFC)|axiom of infinity]] states that there is a [[Sets|set]] containing the empty set and $\{ a \}$ for every $a$ that it contains: $\exists a (\emptyset \in a \land x (x\in a \rightarrow \{ x \} \in a))$This is the set of [[Natural Numbers|natural numbers]] denoted $\mathbb{N}.$ Instead of $\mathbb{N} = \{ \emptyset, \{ \emptyset \}, \{ \{ \emptyset \} \}, \{ \{ \{ \emptyset \} \} \}, \dots \}$ we write $\mathbb{N}=\{ 0,1,2,3\dots \}.$ > # Operations These operations can be shown to satisfy the Peano axioms